Causality

@article{trethowan1938causality,
    author = "Trethowan, Illtyd",
    title = "Causality",
    year = "1938",
    journal = "The Downside Review",
    url = "https://doi.org/10.1177/001258063805600103",
    doi = "10.1177/001258063805600103",
    number = "1",
    pages = "18-30",
    volume = "56"
}

It is usually assumed that space-time is a continuum. This assumption is not required by Lorentz invariance. In this paper we give an example of a Lorentz invariant discrete space-time.Received 13 May 1946DOI:https://doi.org/10.1103/PhysRev.71.38©1947 American Physical Society

@article{doi101103physrev7138,
    author = "Snyder, Hartland S.",
    title = "Quantized Space-Time",
    year = "1947",
    journal = "Physical Review",
    abstract = "It is usually assumed that space-time is a continuum. This assumption is not required by Lorentz invariance. In this paper we give an example of a Lorentz invariant discrete space-time.Received 13 May 1946DOI:https://doi.org/10.1103/PhysRev.71.38©1947 American Physical Society",
    url = "https://doi.org/10.1103/physrev.71.38",
    doi = "10.1103/physrev.71.38",
    openalex = "W2225001832"
}

In this paper two things are done. (1) It is shown that a considerable simplification can be attained in writing down matrix elements for complex processes in electrodynamics. Further, a physical point of view is available which permits them to be written down directly for any specific problem. Being simply a restatement of conventional electrodynamics, however, the matrix elements diverge for complex processes. (2) Electrodynamics is modified by altering the interaction of electrons at short distances. All matrix elements are now finite, with the exception of those relating to problems of vacuum polarization. The latter are evaluated in a manner suggested by Pauli and Bethe, which gives finite results for these matrices also. The only effects sensitive to the modification are changes in mass and charge of the electrons. Such changes could not be directly observed. Phenomena directly observable, are insensitive to the details of the modification used (except at extreme energies). For such phenomena, a limit can be taken as the range of the modification goes to zero. The results then agree with those of Schwinger. A complete, unambiguous, and presumably consistent, method is therefore available for the calculation of all processes involving electrons and photons.The simplification in writing the expressions results from an emphasis on the over-all space-time view resulting from a study of the solution of the equations of electrodynamics. The relation of this to the more conventional Hamiltonian point of view is discussed. It would be very difficult to make the modification which is proposed if one insisted on having the equations in Hamiltonian form.The methods apply as well to charges obeying the Klein-Gordon equation, and to the various meson theories of nuclear forces. Illustrative examples are given. Although a modification like that used in electrodynamics can make all matrices finite for all of the meson theories, for some of the theories it is no longer true that all directly observable phenomena are insensitive to the details of the modification used.The actual evaluation of integrals appearing in the matrix elements may be facilitated, in the simpler cases, by methods described in the appendix.

@article{doi101103physrev76769,
    author = "Feynman, Richard P.",
    title = "Space-Time Approach to Quantum Electrodynamics",
    year = "1949",
    journal = "Physical Review",
    abstract = "In this paper two things are done. (1) It is shown that a considerable simplification can be attained in writing down matrix elements for complex processes in electrodynamics. Further, a physical point of view is available which permits them to be written down directly for any specific problem. Being simply a restatement of conventional electrodynamics, however, the matrix elements diverge for complex processes. (2) Electrodynamics is modified by altering the interaction of electrons at short distances. All matrix elements are now finite, with the exception of those relating to problems of vacuum polarization. The latter are evaluated in a manner suggested by Pauli and Bethe, which gives finite results for these matrices also. The only effects sensitive to the modification are changes in mass and charge of the electrons. Such changes could not be directly observed. Phenomena directly observable, are insensitive to the details of the modification used (except at extreme energies). For such phenomena, a limit can be taken as the range of the modification goes to zero. The results then agree with those of Schwinger. A complete, unambiguous, and presumably consistent, method is therefore available for the calculation of all processes involving electrons and photons.The simplification in writing the expressions results from an emphasis on the over-all space-time view resulting from a study of the solution of the equations of electrodynamics. The relation of this to the more conventional Hamiltonian point of view is discussed. It would be very difficult to make the modification which is proposed if one insisted on having the equations in Hamiltonian form.The methods apply as well to charges obeying the Klein-Gordon equation, and to the various meson theories of nuclear forces. Illustrative examples are given. Although a modification like that used in electrodynamics can make all matrices finite for all of the meson theories, for some of the theories it is no longer true that all directly observable phenomena are insensitive to the details of the modification used.The actual evaluation of integrals appearing in the matrix elements may be facilitated, in the simpler cases, by methods described in the appendix.",
    url = "https://doi.org/10.1103/physrev.76.769",
    doi = "10.1103/physrev.76.769",
    openalex = "W2081796865"
}

The general aim is to obtain maximum information about the $S$-matrix with a minimum of assumptions concerning the interaction. This program is carried through for the scattering of the electromagnetic field by a fixed center. The center is assumed spherically symmetric and of finite size, so that the causality condition can be applied. From this condition it follows rigorously that the $S$-matrix has a one-valued analytic continuation, whose only singularities are poles in the lower half-plane, and whose behavior at infinity can be specified. Particular consequences are: (i) the analytic properties of Wigner's function $R$; (ii) the integral relation connecting real and imaginary parts of $S$; (iii) relations connecting the sum of the oscillator strengths with the scattering cross section.

@article{doi101103physrev891072,
    author = "van Kampen, N. G.",
    title = "S -Matrix and Causality Condition. I. Maxwell Field",
    year = "1953",
    journal = "Physical Review",
    abstract = "The general aim is to obtain maximum information about the $S$-matrix with a minimum of assumptions concerning the interaction. This program is carried through for the scattering of the electromagnetic field by a fixed center. The center is assumed spherically symmetric and of finite size, so that the causality condition can be applied. From this condition it follows rigorously that the $S$-matrix has a one-valued analytic continuation, whose only singularities are poles in the lower half-plane, and whose behavior at infinity can be specified. Particular consequences are: (i) the analytic properties of Wigner's function $R$; (ii) the integral relation connecting real and imaginary parts of $S$; (iii) relations connecting the sum of the oscillator strengths with the scattering cross section.",
    url = "https://doi.org/10.1103/physrev.89.1072",
    doi = "10.1103/physrev.89.1072",
    openalex = "W2051953758"
}

The application of the "causality condition" to the $S$ matrix for nonrelativistic particles encounters several difficulties: (a) there is no maximum velocity; (b) the interference of ingoing and outgoing waves has to be taken into account; (c) wave packets with a sharp front do not exist. The condition is therefore reformulated as follows: At any time the total probability of finding the particle outside the scattering center shall not be greater than 1, for every form of the incident wave packet. From this follows for spherical waves that $S$, as a function of the momentum $p$, is analytic and holomorphic in the first quadrant and that ${e}^{2iap}S(p)$ (where $a$ is the radius of the scattering center) has an imaginary part \ensuremath{\leqslant} 1. That suffices to give an explicit integral representation and a product expansion for $S$, but these permit a more general form for $S$ than is usually envisaged. If, however, the usual symmetry relation $S(\ensuremath{-}p)=S{(p)}^{*}$ is assumed in addition to the causality condition, more specific equations can be derived, which are direct generalizations of those in Part I. In particular, integral relations between the real and imaginary parts of $S$, and the properties that Wigner found for the $R$ matrix can be deduced.

@article{doi101103physrev911267,
    author = "van Kampen, N. G.",
    title = "S Matrix and Causality Condition. II. Nonrelativistic Particles",
    year = "1953",
    journal = "Physical Review",
    abstract = {The application of the "causality condition" to the $S$ matrix for nonrelativistic particles encounters several difficulties: (a) there is no maximum velocity; (b) the interference of ingoing and outgoing waves has to be taken into account; (c) wave packets with a sharp front do not exist. The condition is therefore reformulated as follows: At any time the total probability of finding the particle outside the scattering center shall not be greater than 1, for every form of the incident wave packet. From this follows for spherical waves that $S$, as a function of the momentum $p$, is analytic and holomorphic in the first quadrant and that ${e}^{2iap}S(p)$ (where $a$ is the radius of the scattering center) has an imaginary part \ensuremath{\leqslant} 1. That suffices to give an explicit integral representation and a product expansion for $S$, but these permit a more general form for $S$ than is usually envisaged. If, however, the usual symmetry relation $S(\ensuremath{-}p)=S{(p)}^{*}$ is assumed in addition to the causality condition, more specific equations can be derived, which are direct generalizations of those in Part I. In particular, integral relations between the real and imaginary parts of $S$, and the properties that Wigner found for the $R$ matrix can be deduced.},
    url = "https://doi.org/10.1103/physrev.91.1267",
    doi = "10.1103/physrev.91.1267",
    openalex = "W2017461348"
}

The limitations on scattering amplitudes imposed by causality requirements are deduced from the demand that the commutator of field operators vanish if the operators are taken at points with space-like separations. The problems of the scattering of spin-zero particles by a force center and the scattering of photons by a quantized matter field are discussed. The causality requirements lead in a natural way to the well-known dispersion relation of Kramers and Kronig. A new sum rule for the nuclear photoeffect is derived and the scattering of photons by nucleons is discussed.

@article{doi101103physrev951612,
    author = "Gell‐Mann, Murray and Goldberger, M. L. and Thirring, Walter",
    title = "Use of Causality Conditions in Quantum Theory",
    year = "1954",
    journal = "Physical Review",
    abstract = "The limitations on scattering amplitudes imposed by causality requirements are deduced from the demand that the commutator of field operators vanish if the operators are taken at points with space-like separations. The problems of the scattering of spin-zero particles by a force center and the scattering of photons by a quantized matter field are discussed. The causality requirements lead in a natural way to the well-known dispersion relation of Kramers and Kronig. A new sum rule for the nuclear photoeffect is derived and the scattering of photons by nucleons is discussed.",
    url = "https://doi.org/10.1103/physrev.95.1612",
    doi = "10.1103/physrev.95.1612",
    openalex = "W2052995304",
    references = "doi101016s0031891446800788, doi101103physrev75651, doi101103physrev76790, doi101103physrev78115, doi101103physrev81115, doi101103physrev891072, doi101103physrev911267, doi101103physrev93233, doi10111911933407, doi101364josa12000547"
}

"Strict causality" is the assumption that no signal whatsoever can be transmitted over a space-like interval in space-time, or that no signal can travel faster than the velocity of light in vacuo. In this paper a rigorous proof is given of the logical equivalence of strict causality ("no output before the input") and the validity of a dispersion relation, e.g., the relation expressing the real part of a generalized scattering amplitude as an integral involving the imaginary part. This proof applies to a general linear system with a time-independent connection between the output and a freely variable input and has the advantage over previous work that no tacit assumptions are made about the analytic behavior or single-valuedness of the amplitude, but, on the contrary, strict causality is shown to imply that the generalized scattering amplitude is analytic in the upper half of the complex frequency plane. The dispersion relations are given first as a relation between the real and imaginary parts of the generalized scattering amplitude and then in terms of the complex phase shift.

@article{doi101103physrev1041760,
    author = "Toll, John S.",
    title = "Causality and the Dispersion Relation: Logical Foundations",
    year = "1956",
    journal = "Physical Review",
    abstract = {"Strict causality" is the assumption that no signal whatsoever can be transmitted over a space-like interval in space-time, or that no signal can travel faster than the velocity of light in vacuo. In this paper a rigorous proof is given of the logical equivalence of strict causality ("no output before the input") and the validity of a dispersion relation, e.g., the relation expressing the real part of a generalized scattering amplitude as an integral involving the imaginary part. This proof applies to a general linear system with a time-independent connection between the output and a freely variable input and has the advantage over previous work that no tacit assumptions are made about the analytic behavior or single-valuedness of the amplitude, but, on the contrary, strict causality is shown to imply that the generalized scattering amplitude is analytic in the upper half of the complex frequency plane. The dispersion relations are given first as a relation between the real and imaginary parts of the generalized scattering amplitude and then in terms of the complex phase shift.},
    url = "https://doi.org/10.1103/physrev.104.1760",
    doi = "10.1103/physrev.104.1760",
    openalex = "W2076972285",
    references = "doi101103physrev951612"
}

With the introduction of multiply connected topologies into physics, a question of causality arises. There are alternative routes between two points in a multiply connected space. Therefore, one may ask if a signal traveling at the speed of light along one route could be outpaced by a signal which has traveled a much shorter path through a handle or "wormhole." This paper examines one such situation and shows that in this example causality is preserved. It proves essential in the analysis to distinguish between those regions of space-time which are catastrophic and those which are not. A catastrophic region is composed of catastrophic points. A catastrophic point in space-time is so located with respect to eventual singularities in the intrinsic geometry that every time-like geodesic through it necessarily runs into a region of infinite curvature at some time in the future---or was born out of a region of infinite curvature at some time in the past---or both. If a classical analysis of nature were possible---which it is not---then it would be natural to postulate that laboratory physics is carried out in noncatastrophic regions of space-time. Two such regions are shown to exist in the example considered in the paper. It is shown that no signal can ever be sent from one to the other. The key point in preventing any violation of causality is simple: The (Schwarzschild) throat of the wormhole pinches off in a finite time and traps the signal in a region of infinite curvature. This investigation also displays some of the unusual geometric features of the Schwarzschild solution of Einstein's equations for a spherically symmetrical center of attraction. Radial spacelike geodesics passing through the throat are calculated and it is shown that there exist regions of space-time unreachable by any radial geodesics that issue from a given point. Also, there exist points in space-time from which light signals can never be received no matter how long one waits.

@article{doi101103physrev128919,
    author = "Fuller, Robert W. and Wheeler, John",
    title = "Causality and Multiply Connected Space-Time",
    year = "1962",
    journal = "Physical Review",
    abstract = {With the introduction of multiply connected topologies into physics, a question of causality arises. There are alternative routes between two points in a multiply connected space. Therefore, one may ask if a signal traveling at the speed of light along one route could be outpaced by a signal which has traveled a much shorter path through a handle or "wormhole." This paper examines one such situation and shows that in this example causality is preserved. It proves essential in the analysis to distinguish between those regions of space-time which are catastrophic and those which are not. A catastrophic region is composed of catastrophic points. A catastrophic point in space-time is so located with respect to eventual singularities in the intrinsic geometry that every time-like geodesic through it necessarily runs into a region of infinite curvature at some time in the future---or was born out of a region of infinite curvature at some time in the past---or both. If a classical analysis of nature were possible---which it is not---then it would be natural to postulate that laboratory physics is carried out in noncatastrophic regions of space-time. Two such regions are shown to exist in the example considered in the paper. It is shown that no signal can ever be sent from one to the other. The key point in preventing any violation of causality is simple: The (Schwarzschild) throat of the wormhole pinches off in a finite time and traps the signal in a region of infinite curvature. This investigation also displays some of the unusual geometric features of the Schwarzschild solution of Einstein's equations for a spherically symmetrical center of attraction. Radial spacelike geodesics passing through the throat are calculated and it is shown that there exist regions of space-time unreachable by any radial geodesics that issue from a given point. Also, there exist points in space-time from which light signals can never be received no matter how long one waits.},
    url = "https://doi.org/10.1103/physrev.128.919",
    doi = "10.1103/physrev.128.919",
    openalex = "W2090749997"
}

Causality is represented by a partial ordering on Minkowski space, and the group of all automorphisms that preserve this partial ordering is shown to be generated by the inhomogeneous Lorentz group and dilatations.

@article{doi10106311704140,
    author = "Zeeman, E. C.",
    title = "Causality Implies the Lorentz Group",
    year = "1964",
    journal = "Journal of Mathematical Physics",
    abstract = "Causality is represented by a partial ordering on Minkowski space, and the group of all automorphisms that preserve this partial ordering is shown to be generated by the inhomogeneous Lorentz group and dilatations.",
    url = "https://doi.org/10.1063/1.1704140",
    doi = "10.1063/1.1704140",
    openalex = "W2027332450"
}

It is known that the entire geometry of many relativistic space-times can be summed up in two concepts, a space-time measure $\ensuremath{\mu}$ and a space-time causal or chronological order relation $C$, defining a causal measure space. On grounds of finiteness, unity, and symmetry, we argue that macroscopic space-time may be the classical-geometrical limit of a causal quantum space. A tentative conceptual framework is provided. Mathematical individuals that naturally form causal spaces are symbol sets or words, taken in the order of their generation. The natural extension of this purely logical concept to quantum symbols is formulated. The problem is posed of giving finite quantum rules for the generation of quantum symbol sets such that the order of generation becomes, in the classical limit, the causal order of space-time---as it were, to break the space-time code. The causal quantum spaces of three simple codes are generated for comparison with reality. The singulary code (repetitions of one digit) gives a linearly ordered external world of one time dimension and a circular internal space. The binary code gives the future null cone of special relativity and a circular internal space. The causal quantum space of word pairs in the binary code gives the solid light cone $t>{({x}^{2}+{y}^{2}+{z}^{2})}^{\frac{1}{2}}$ of special relativity and an internal space $U(2,C)$ suitable for the description of charge and isospin. In the classical limit, there is full translational and proper Lorentz invariance except at the boundary of the light cone, where the classical-geometrical limit fails. Plausible consequences of this model for cosmology and elementary particles are discussed. There is a quantum of time $\ensuremath{\tau}\ensuremath{\lesssim}\frac{\ensuremath{\hbar}}{{m}_{\ensuremath{\mu}}{c}^{3}}$ and a space-time complementarity relation $\ensuremath{\Delta}t\ensuremath{\Delta}x\ensuremath{\Delta}y\ensuremath{\Delta}z\ensuremath{\gtrsim}{\ensuremath{\tau}}^{4}$.

@article{doi101103physrev1841261,
    author = "Finkelstein, David",
    title = "Space-Time Code",
    year = "1969",
    journal = "Physical Review",
    abstract = "It is known that the entire geometry of many relativistic space-times can be summed up in two concepts, a space-time measure $\ensuremath{\mu}$ and a space-time causal or chronological order relation $C$, defining a causal measure space. On grounds of finiteness, unity, and symmetry, we argue that macroscopic space-time may be the classical-geometrical limit of a causal quantum space. A tentative conceptual framework is provided. Mathematical individuals that naturally form causal spaces are symbol sets or words, taken in the order of their generation. The natural extension of this purely logical concept to quantum symbols is formulated. The problem is posed of giving finite quantum rules for the generation of quantum symbol sets such that the order of generation becomes, in the classical limit, the causal order of space-time---as it were, to break the space-time code. The causal quantum spaces of three simple codes are generated for comparison with reality. The singulary code (repetitions of one digit) gives a linearly ordered external world of one time dimension and a circular internal space. The binary code gives the future null cone of special relativity and a circular internal space. The causal quantum space of word pairs in the binary code gives the solid light cone $t>{({x}^{2}+{y}^{2}+{z}^{2})}^{\frac{1}{2}}$ of special relativity and an internal space $U(2,C)$ suitable for the description of charge and isospin. In the classical limit, there is full translational and proper Lorentz invariance except at the boundary of the light cone, where the classical-geometrical limit fails. Plausible consequences of this model for cosmology and elementary particles are discussed. There is a quantum of time $\ensuremath{\tau}\ensuremath{\lesssim}\frac{\ensuremath{\hbar}}{{m}\_{\ensuremath{\mu}}{c}^{3}}$ and a space-time complementarity relation $\ensuremath{\Delta}t\ensuremath{\Delta}x\ensuremath{\Delta}y\ensuremath{\Delta}z\ensuremath{\gtrsim}{\ensuremath{\tau}}^{4}$.",
    url = "https://doi.org/10.1103/physrev.184.1261",
    doi = "10.1103/physrev.184.1261",
    openalex = "W2055748416",
    references = "doi101017s030500410004144x, doi10106311704140, doi101103physrev7138, doi10111911975143, doi1032917hmj1558576822, doi1043249780203203866"
}

Einstein's General Theory of Relativity leads to two remarkable predictions: first, that the ultimate destiny of many massive stars is to undergo gravitational collapse and to disappear from view, leaving behind a 'black hole' in space; and secondly, that there will exist singularities in space-time itself. These singularities are places where space-time begins or ends, and the presently known laws of physics break down. They will occur inside black holes, and in the past are what might be construed as the beginning of the universe. To show how these predictions arise, the authors discuss the General Theory of Relativity in the large. Starting with a precise formulation of the theory and an account of the necessary background of differential geometry, the significance of space-time curvature is discussed and the global properties of a number of exact solutions of Einstein's field equations are examined. The theory of the causal structure of a general space-time is developed, and is used to study black holes and to prove a number of theorems establishing the inevitability of singualarities under certain conditions. A discussion of the Cauchy problem for General Relativity is also included in this 1973 book.

@book{doi101017cbo9780511524646,
    author = "Hawking, S. W. and Ellis, George",
    title = "The Large Scale Structure of Space-Time",
    year = "1973",
    booktitle = "Cambridge University Press eBooks",
    abstract = "Einstein's General Theory of Relativity leads to two remarkable predictions: first, that the ultimate destiny of many massive stars is to undergo gravitational collapse and to disappear from view, leaving behind a 'black hole' in space; and secondly, that there will exist singularities in space-time itself. These singularities are places where space-time begins or ends, and the presently known laws of physics break down. They will occur inside black holes, and in the past are what might be construed as the beginning of the universe. To show how these predictions arise, the authors discuss the General Theory of Relativity in the large. Starting with a precise formulation of the theory and an account of the necessary background of differential geometry, the significance of space-time curvature is discussed and the global properties of a number of exact solutions of Einstein's field equations are examined. The theory of the causal structure of a general space-time is developed, and is used to study black holes and to prove a number of theorems establishing the inevitability of singualarities under certain conditions. A discussion of the Cauchy problem for General Relativity is also included in this 1973 book.",
    url = "https://doi.org/10.1017/cbo9780511524646",
    doi = "10.1017/cbo9780511524646",
    openalex = "W2029403139"
}

The concept of a quantum dynamics is recapitulated. The Dirac equation is obtained from a pure quantum dynamics as the limit of classical time. The theory is defective in projective gauge invariance and semantic consistency, but illustrates the relation between dynamical and experimental elements of q dynamics, and is finite, Lorentz-invariant, and local.

@article{doi101103physrevd92231,
    author = "Finkelstein, David and Frye, Graham and Susskind, Leonard",
    title = "Space-time code. V",
    year = "1974",
    journal = "Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields",
    abstract = "The concept of a quantum dynamics is recapitulated. The Dirac equation is obtained from a pure quantum dynamics as the limit of classical time. The theory is defective in projective gauge invariance and semantic consistency, but illustrates the relation between dynamical and experimental elements of q dynamics, and is finite, Lorentz-invariant, and local.",
    url = "https://doi.org/10.1103/physrevd.9.2231",
    doi = "10.1103/physrevd.9.2231",
    openalex = "W2111346223"
}

A new topology is proposed for strongly causal space–times. Unlike the standard manifold topology (which merely characterizes continuity properties), the new topology determines the causal, differential, and conformal structures of space–time. The topology is more appealing, physical, and manageable than the topology previously proposed by Zeeman for Minkowski space. It thus seems that many calculations involving the above structures may be made purely topological.

@article{doi1010631522874,
    author = "Hawking, S. W. and King, Alannah and McCarthy, Patrick J.",
    title = "A new topology for curved space–time which incorporates the causal, differential, and conformal structures",
    year = "1976",
    journal = "Journal of Mathematical Physics",
    abstract = "A new topology is proposed for strongly causal space–times. Unlike the standard manifold topology (which merely characterizes continuity properties), the new topology determines the causal, differential, and conformal structures of space–time. The topology is more appealing, physical, and manageable than the topology previously proposed by Zeeman for Minkowski space. It thus seems that many calculations involving the above structures may be made purely topological.",
    url = "https://doi.org/10.1063/1.522874",
    doi = "10.1063/1.522874",
    openalex = "W2157084252"
}

This volume introduces and systematically develops the calculus of 2-spinors. This is the first detailed exposition of this technique which leads not only to a deeper understanding of the structure of space-time, but also provides shortcuts to some very tedious calculations. Many results are given here for the first time.

@book{doi101017cbo9780511564048,
    author = "Penrose, Roger and Rindler, Wolfgang",
    title = "Spinors and Space-Time",
    year = "1984",
    booktitle = "Cambridge University Press eBooks",
    abstract = "This volume introduces and systematically develops the calculus of 2-spinors. This is the first detailed exposition of this technique which leads not only to a deeper understanding of the structure of space-time, but also provides shortcuts to some very tedious calculations. Many results are given here for the first time.",
    url = "https://doi.org/10.1017/cbo9780511564048",
    doi = "10.1017/cbo9780511564048",
    openalex = "W1789645782"
}

@misc{morris1984times1,
    author = "Morris, R",
    title = "Time's Arrows",
    year = "1984",
    howpublished = "New York, Simon and Schuster",
    note = "talkorigins\_source = {true}; raw\_reference = {Morris, R., 1984, Time's Arrows: New York, Simon and Schuster.}"
}

The finite-element (collocation) method enables us to construct discrete time-lattice quantum systems that accurately approximate continuum quantum systems. The discrete quantum systems so generated are fully consistent quantum-mechanical systems in their own right. This paper gives a comprehensive treatment of such quantum systems. We examine various finite-element schemes, construct the effective lattice Hamiltonian, and calculate eigenvalues. Numerical results are extremely easy to obtain and are very accurate.

@article{doi101103physrevd321476,
    author = "Bender, Carl M. and Milton, Kimball A. and Sharp, David H. and Simmons, L. M. and Stong, Richard",
    title = "Discrete-time quantum mechanics",
    year = "1985",
    journal = "Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields",
    abstract = "The finite-element (collocation) method enables us to construct discrete time-lattice quantum systems that accurately approximate continuum quantum systems. The discrete quantum systems so generated are fully consistent quantum-mechanical systems in their own right. This paper gives a comprehensive treatment of such quantum systems. We examine various finite-element schemes, construct the effective lattice Hamiltonian, and calculate eigenvalues. Numerical results are extremely easy to obtain and are very accurate.",
    url = "https://doi.org/10.1103/physrevd.32.1476",
    doi = "10.1103/physrevd.32.1476",
    openalex = "W2054343861"
}

The usual proof of the CPT theorem does not apply to theories which include the gravitational field. Nevertheless, it is shown that CPT invariance still holds in these cases provided that, as has recently been proposed, the quantum state of the Universe is defined by a path integral over metrics that are compact without boundary. The observed asymmetry or arrow of time defined by the direction of time in which entropy increases is shown to be related to the cosmological arrow of time defined by the direction of time in which the Universe is expanding. It arises because in the proposed quantum state the Universe would have been smooth and homogeneous when it was small but irregular and inhomogeneous when it was large. The thermodynamic arrow would reverse during a contracting phase of the Universe or inside black holes. Possible observational tests of this prediction are discussed.

@article{doi101103physrevd322489,
    author = "Hawking, S. W.",
    title = "Arrow of time in cosmology",
    year = "1985",
    journal = "Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields",
    abstract = "The usual proof of the CPT theorem does not apply to theories which include the gravitational field. Nevertheless, it is shown that CPT invariance still holds in these cases provided that, as has recently been proposed, the quantum state of the Universe is defined by a path integral over metrics that are compact without boundary. The observed asymmetry or arrow of time defined by the direction of time in which entropy increases is shown to be related to the cosmological arrow of time defined by the direction of time in which the Universe is expanding. It arises because in the proposed quantum state the Universe would have been smooth and homogeneous when it was small but irregular and inhomogeneous when it was large. The thermodynamic arrow would reverse during a contracting phase of the Universe or inside black holes. Possible observational tests of this prediction are discussed.",
    url = "https://doi.org/10.1103/physrevd.32.2489",
    doi = "10.1103/physrevd.32.2489",
    openalex = "W1998523925",
    references = "doi101007bf01206031, doi101016055032137890161x, doi1010160550321384900932, doi101017cbo9780511524646, doi10108800319112264026, doi10108800319112314029, doi101103physrevd282960, doi101103physrevd311777, doi101103revmodphys17157, doi101103revmodphys21425"
}

We propose that space-time at the smallest scales is in reality a causal set: a locally finite set of elements endowed with a partial order corresponding to the macroscopic relation that defines past and future. We explore how a Lorentzian manifold can approximate a causal set, noting in particular that the thereby defined effective dimensionality of a given causal set can vary with length scale. Finally, we speculate briefly on the quantum dynamics of causal sets, indicating why an appropriate choice of action can reproduce general relativity in the classical limit.

@article{doi101103physrevlett59521,
    author = "Bombelli, L. and Lee, Joohan and Meyer, David and Sorkin, Rafael D.",
    title = "Space-time as a causal set",
    year = "1987",
    journal = "Physical Review Letters",
    abstract = "We propose that space-time at the smallest scales is in reality a causal set: a locally finite set of elements endowed with a partial order corresponding to the macroscopic relation that defines past and future. We explore how a Lorentzian manifold can approximate a causal set, noting in particular that the thereby defined effective dimensionality of a given causal set can vary with length scale. Finally, we speculate briefly on the quantum dynamics of causal sets, indicating why an appropriate choice of action can reproduce general relativity in the classical limit.",
    url = "https://doi.org/10.1103/physrevlett.59.521",
    doi = "10.1103/physrevlett.59.521",
    openalex = "W2077470755",
    references = "doi1010160370269377906785, doi1010160370269386914255, doi1010160550321385905061, doi101017s030500410004144x, doi1010631522874, doi1010631523436, doi101103physrev1841261, doi101103physrevd16953, doi101103physrevd323201, doi101103physrevlett551846"
}

The concept of time is introduced as a major topic for organizational and management research. Including a discussion of differing times and temporalities, macro level research and theory are described that relate time to such substantive areas as organizational culture, strategic planning, and organizational contingency theory. At the micro level, theory and research on time and individual differences, decision making, motivation, and group behavior are reviewed critically. Organizational and management topics of particular salience forfuture temporal research and management practice are identified.

@article{doi101177014920638801400209,
    author = "Bluedorn, Allen C. and Denhardt, Robert B.",
    title = "Time and Organizations",
    year = "1988",
    journal = "Journal of Management",
    abstract = "The concept of time is introduced as a major topic for organizational and management research. Including a discussion of differing times and temporalities, macro level research and theory are described that relate time to such substantive areas as organizational culture, strategic planning, and organizational contingency theory. At the micro level, theory and research on time and individual differences, decision making, motivation, and group behavior are reviewed critically. Organizational and management topics of particular salience forfuture temporal research and management practice are identified.",
    url = "https://doi.org/10.1177/014920638801400209",
    doi = "10.1177/014920638801400209",
    openalex = "W2142325015",
    references = "openalexw3021036590"
}

We argue that the Wheeler-DeWitt equation with a consistent boundary condition is only compatible with an arrow of time that formally reverses in a recollapsing universe. To recover a classically recollapsing universe in terms of wave packets, we impose the usual boundary condition of excluding exponentially increasing wave functions for large scale factors. Consistency of these opposite arrows is then facilitated by quantum effects in the region of the classical turning point. We also discuss in this context the meaning of the time-asymmetric expression used in the definition of ``consistent histories.'' Since gravitational time dilation diverges at horizons, one has to conclude that collapsing matter must start reexpanding ``anticausally'' (controlled by the reversed arrow) in this scenario before horizons or singularities can form. There would then also be no mass inflation nor any information loss paradox.

@article{doi101103physrevd514145,
    author = "Kiefer, Claus and Zeh, H. D.",
    title = "Arrow of time in a recollapsing quantum universe",
    year = "1995",
    journal = "Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields",
    abstract = "We argue that the Wheeler-DeWitt equation with a consistent boundary condition is only compatible with an arrow of time that formally reverses in a recollapsing universe. To recover a classically recollapsing universe in terms of wave packets, we impose the usual boundary condition of excluding exponentially increasing wave functions for large scale factors. Consistency of these opposite arrows is then facilitated by quantum effects in the region of the classical turning point. We also discuss in this context the meaning of the time-asymmetric expression used in the definition of ``consistent histories.'' Since gravitational time dilation diverges at horizons, one has to conclude that collapsing matter must start reexpanding ``anticausally'' (controlled by the reversed arrow) in this scenario before horizons or singularities can form. There would then also be no mass inflation nor any information loss paradox.",
    url = "https://doi.org/10.1103/physrevd.51.4145",
    doi = "10.1103/physrevd.51.4145",
    openalex = "W1985544149",
    references = "doi101103physrevd322489"
}

I suggest in this paper a new strategy to attack the problem of the reality conditions in the Ashtekar approach to classical and quantum general relativity. By writing a modified Hamiltonian constraint in the usual SO(3) Yang-Mills phase space I show that it is possible to describe space-times with a Lorentzian signature without the introduction of complex variables. All the features of the Ashtekar formalism related to the geometrical nature of the new variables are retained; in particular, it is still possible, in principle, to use the loop variables approach in the passage to the quantum theory. The key issue in the new formulation is how to deal with the more complicated Hamiltonian constraint that must be used in order to avoid the introduction of complex fields.

@article{doi101103physrevd515507,
    author = "Barbero, J F G",
    title = "Real Ashtekar variables for Lorentzian signature space-times",
    year = "1995",
    journal = "Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields",
    abstract = "I suggest in this paper a new strategy to attack the problem of the reality conditions in the Ashtekar approach to classical and quantum general relativity. By writing a modified Hamiltonian constraint in the usual SO(3) Yang-Mills phase space I show that it is possible to describe space-times with a Lorentzian signature without the introduction of complex variables. All the features of the Ashtekar formalism related to the geometrical nature of the new variables are retained; in particular, it is still possible, in principle, to use the loop variables approach in the passage to the quantum theory. The key issue in the new formulation is how to deal with the more complicated Hamiltonian constraint that must be used in order to avoid the introduction of complex fields.",
    url = "https://doi.org/10.1103/physrevd.51.5507",
    doi = "10.1103/physrevd.51.5507",
    openalex = "W2126106103",
    references = "doi101103physrevd361587, doi101103physrevlett572244"
}

We propose a canonical procedure to quantize fields with interaction on discrete space-time. The time evolution operator that reproduces the field equation is represented by using canonical variables. The generator of the operator is a conserved quantity, but its existence is not obvious. It is possible to calculate the S-matrix perturbatively. Our quantization gives the same results as those given by the path integral quantization. 1. Introduction

@article{doi101143ptp94249,
    author = "Hashimoto, T. and Hayashi, A. and Horibe, M. and Yamamoto, H.",
    title = "How to Quantize Fields Canonically on Discrete Space-Time",
    year = "1995",
    journal = "Progress of Theoretical Physics",
    abstract = "We propose a canonical procedure to quantize fields with interaction on discrete space-time. The time evolution operator that reproduces the field equation is represented by using canonical variables. The generator of the operator is a conserved quantity, but its existence is not obvious. It is possible to calculate the S-matrix perturbatively. Our quantization gives the same results as those given by the path integral quantization. 1. Introduction",
    url = "https://doi.org/10.1143/ptp.94.249",
    doi = "10.1143/ptp.94.249",
    openalex = "W2097736033"
}

We discuss the principles to be used in the construction of discrete time classical and quantum mechanics as applied to point particle systems. In the classical theory this includes the concept of virtual path and the construction of system functions from classical Lagrangians, Cadzow’s variational principle applied to the action sum, Maeda-Noether and Logan invariants of the motion, elliptic and hyperbolic harmonic oscillator behaviour, gauge invariant electrodynamics and charge conservation, and the Grassmannian oscillator. First quantised discrete time mechanics is discussed via the concept of system amplitude, which permits the construction of all quantities of interest such as commutators and scattering amplitudes. We discuss stroboscopic quantum mechanics, or the construction of discrete time quantum theory from continuous time quantum theory and show how this works in detail for the free Newtonian particle. We conclude with an application of the Schwinger action principle to the important case of the quantised discrete time inhomogeneous oscillator. 1

@article{openalexw3125786388,
    author = "Jaroszkiewicz, George and Norton, Keith",
    title = "Principles of Discrete Time Mechanics: I. Particle Systems",
    year = "1996",
    abstract = "We discuss the principles to be used in the construction of discrete time classical and quantum mechanics as applied to point particle systems. In the classical theory this includes the concept of virtual path and the construction of system functions from classical Lagrangians, Cadzow’s variational principle applied to the action sum, Maeda-Noether and Logan invariants of the motion, elliptic and hyperbolic harmonic oscillator behaviour, gauge invariant electrodynamics and charge conservation, and the Grassmannian oscillator. First quantised discrete time mechanics is discussed via the concept of system amplitude, which permits the construction of all quantities of interest such as commutators and scattering amplitudes. We discuss stroboscopic quantum mechanics, or the construction of discrete time quantum theory from continuous time quantum theory and show how this works in detail for the free Newtonian particle. We conclude with an application of the Schwinger action principle to the important case of the quantised discrete time inhomogeneous oscillator. 1",
    openalex = "W3125786388",
    references = "doi101007bf01832628, doi1010160370269383906871, doi10108000207177008905922, doi10108803054470264023, doi10108803054470309023, doi101103physrevd321476, doi101143ptp931173, doi101143ptp94249, openalexw3104223222"
}

We apply the principles discussed in an earlier paper to the construction of discrete time field theories. We derive the discrete time field equations of motion and Noether's theorem and apply them to the Schrodinger equation to illustrate the methodology. Stationary solutions to the discrete time Schrodinger wave equation are found to be identical to standard energy eigenvalue solutions except for a fundamental limit on the energy. Then we apply the formalism to the free neutral Klein Gordon system, deriving the equations of motion and conserved quantities such as the linear momentum and angular momentum. We show that there is an upper bound on the magnitude of linear momentum for physical particle-like solutions. We extend the formalism to the charged scalar field coupled to Maxwell's electrodynamics in a gauge invariant way. We apply the formalism to include the Maxwell and Dirac fields, setting the scene for second quantisation of discrete time mechanics and discrete time Quantum Electrodynamics.

@article{doi10108803054470309023,
    author = "Jaroszkiewicz, George and Norton, Keith",
    title = "Principles of discrete time mechanics: II. Classical field theory",
    year = "1997",
    journal = "Journal of Physics A Mathematical and General",
    abstract = "We apply the principles discussed in an earlier paper to the construction of discrete time field theories. We derive the discrete time field equations of motion and Noether's theorem and apply them to the Schrodinger equation to illustrate the methodology. Stationary solutions to the discrete time Schrodinger wave equation are found to be identical to standard energy eigenvalue solutions except for a fundamental limit on the energy. Then we apply the formalism to the free neutral Klein Gordon system, deriving the equations of motion and conserved quantities such as the linear momentum and angular momentum. We show that there is an upper bound on the magnitude of linear momentum for physical particle-like solutions. We extend the formalism to the charged scalar field coupled to Maxwell's electrodynamics in a gauge invariant way. We apply the formalism to include the Maxwell and Dirac fields, setting the scene for second quantisation of discrete time mechanics and discrete time Quantum Electrodynamics.",
    url = "https://doi.org/10.1088/0305-4470/30/9/023",
    doi = "10.1088/0305-4470/30/9/023",
    openalex = "W2051206171"
}

We apply the principles discussed in an earlier paper to the construction of discrete time field theories. We derive the discrete time field equations of motion and Noether’s theorem and apply them to the Schrödinger equation to illustrate the methodology. Stationary solutions to the discrete time Schrödinger wave equation are found to be identical to standard energy eigenvalue solutions except for a fundamental limit on the energy. Then we apply the formalism to the free neutral Klein Gordon system, deriving the equations of motion and conserved quantities such as the linear momentum and angular momentum. We show that there is an upper bound on the magnitude of linear momentum for physical particle-like solutions. We extend the formalism to the charged scalar field coupled to Maxwell’s electrodynamics in a gauge invariant way. We apply the formalism to include the Maxwell and Dirac fields, setting the scene for second quantisation of discrete time mechanics and discrete time Quantum Electrodynamics. 1

@article{openalexw3104223222,
    author = "Jaroszkiewicz, George and Norton, Keith",
    title = "Principles of discrete time mechanics: II. Classical field theory",
    year = "1997",
    abstract = "We apply the principles discussed in an earlier paper to the construction of discrete time field theories. We derive the discrete time field equations of motion and Noether’s theorem and apply them to the Schrödinger equation to illustrate the methodology. Stationary solutions to the discrete time Schrödinger wave equation are found to be identical to standard energy eigenvalue solutions except for a fundamental limit on the energy. Then we apply the formalism to the free neutral Klein Gordon system, deriving the equations of motion and conserved quantities such as the linear momentum and angular momentum. We show that there is an upper bound on the magnitude of linear momentum for physical particle-like solutions. We extend the formalism to the charged scalar field coupled to Maxwell’s electrodynamics in a gauge invariant way. We apply the formalism to include the Maxwell and Dirac fields, setting the scene for second quantisation of discrete time mechanics and discrete time Quantum Electrodynamics. 1",
    openalex = "W3104223222"
}

@article{doi101016s0370269399002671,
    author = "Veneziano, G.",
    title = "Pre-bangian origin of our entropy and time arrow",
    year = "1999",
    journal = "Physics Letters B",
    url = "https://doi.org/10.1016/s0370-2693(99)00267-1",
    doi = "10.1016/s0370-2693(99)00267-1",
    openalex = "W2125534773",
    references = "doi101007bf00670342"
}

While there is much `time-related research', there is little `research on time'. This is striking since time is a key point in understanding organizations, their actions, culture, efficacy, etc. Most studies of time in management and organizational theory take time for granted. While there are numerous studies that address temporal issues, they are widely dispersed and unsystematic. This paper provides a classification of temporal studies of organizations and management. The scheme is built around two criteria: concepts of time and the role of time in research design. In the former, there are two contrasting concepts of time: clock time and social time. In the latter, time plays the roles of independent or dependent variables. By intersecting the two criteria, four notions of temporality (`deciding', `working', `varying' and `changing' times) are introduced to account for a variety of studies of time. The resulting classification not only reveals the current situation of studies about time, but it also indicates a direction which further research effort should take. We conclude by showing that temporally sensitive approaches will benefit research on organizations.

@article{doi1011770170840699206006,
    author = "Lee, Heejin and Liebenau, Jonathan",
    title = "Time in Organizational Studies: Towards a New Research Direction",
    year = "1999",
    journal = "Organization Studies",
    abstract = "While there is much `time-related research', there is little `research on time'. This is striking since time is a key point in understanding organizations, their actions, culture, efficacy, etc. Most studies of time in management and organizational theory take time for granted. While there are numerous studies that address temporal issues, they are widely dispersed and unsystematic. This paper provides a classification of temporal studies of organizations and management. The scheme is built around two criteria: concepts of time and the role of time in research design. In the former, there are two contrasting concepts of time: clock time and social time. In the latter, time plays the roles of independent or dependent variables. By intersecting the two criteria, four notions of temporality (`deciding', `working', `varying' and `changing' times) are introduced to account for a variety of studies of time. The resulting classification not only reveals the current situation of studies about time, but it also indicates a direction which further research effort should take. We conclude by showing that temporally sensitive approaches will benefit research on organizations.",
    url = "https://doi.org/10.1177/0170840699206006",
    doi = "10.1177/0170840699206006",
    openalex = "W2122121725",
    references = "openalexw3021036590"
}

A mathematical definition of classical causality over discrete spacetime dynamics is formulated. The approach is background free and permits a definition of causality in a precise way whenever the spacetime dynamics permits. It gives a natural meaning to the concepts of cosmic time, spacelike hypersurfaces and timelike or lightlike flows without assuming the notion of a background metric. The concepts of causal propagators and the speed of causality are discussed. In this approach the concepts of spacetime and dynamics are linked in an essential and inseparable whole, with no meaning to either on its own.

@article{doi101007978146154285813,
    author = "Jaroszkiewicz, George",
    title = "Discrete spacetime: classical causality,prediction, retrodiction and the mathematical arrow of time",
    year = "2000",
    journal = "CERN Bulletin",
    abstract = "A mathematical definition of classical causality over discrete spacetime dynamics is formulated. The approach is background free and permits a definition of causality in a precise way whenever the spacetime dynamics permits. It gives a natural meaning to the concepts of cosmic time, spacelike hypersurfaces and timelike or lightlike flows without assuming the notion of a background metric. The concepts of causal propagators and the speed of causality are discussed. In this approach the concepts of spacetime and dynamics are linked in an essential and inseparable whole, with no meaning to either on its own.",
    url = "https://doi.org/10.1007/978-1-4615-4285-8\_13",
    doi = "10.1007/978-1-4615-4285-8\_13",
    openalex = "W1620972594",
    references = "doi101017cbo9780511565748, doi101017cbo9780511612909003, doi101103physrev1841261, doi101103physrevd525743, doi101103physrevd92231, doi101103physrevlett59521, openalexw2962925375, openalexw3021036590, openalexw3101093997, openalexw3125786388"
}

@article{doi101016s1355219899000301,
    author = "Rohrlich, F.",
    title = "Causality and the Arrow of Classical Time",
    year = "2000",
    journal = "Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern Physics",
    url = "https://doi.org/10.1016/s1355-2198(99)00030-1",
    doi = "10.1016/s1355-2198(99)00030-1",
    openalex = "W2051180531",
    references = "doi1010079783662025956, doi10100797894015344754, doi1010160003491660900300, doi101017cbo9780511624933, doi10106313059791, doi101098rspa19380124, doi101103physrevd553457, doi101103physrevd563381, doi1023072215819, openalexw1616093546"
}

We discuss the emergence of time dilation as a normal feature expected of any system where a central processor may have to wait one or more clock cycles before concluding a local calculation. We show how the process of causal implication in a typical Newtonian cellular automaton leads naturally to Lorentz transformations and invariant causal structure.

@misc{doi1048550arxivgrqc0008022,
    author = "Jaroszkiewicz, George",
    title = "Causal Implication and the Origin of Time Dilation",
    year = "2000",
    booktitle = "arXiv (Cornell University)",
    abstract = "We discuss the emergence of time dilation as a normal feature expected of any system where a central processor may have to wait one or more clock cycles before concluding a local calculation. We show how the process of causal implication in a typical Newtonian cellular automaton leads naturally to Lorentz transformations and invariant causal structure.",
    url = "https://doi.org/10.48550/arxiv.gr-qc/0008022",
    doi = "10.48550/arxiv.gr-qc/0008022",
    openalex = "W1569125338",
    references = "doi101007978146154285813"
}

The End of Time: The Next Revolution in Physics by Julian Barbour Richard Feynman once quipped that is what happens when nothing else does. But Julian Barbour disagrees: if nothing happened, if nothing changed, then time would stop. For time is nothing but change. It is change that we perceive occurring all around us, not time. Put simply, time does not exist. In this highly provocative volume, Barbour presents the basic evidence for a timeless universe, and shows why we still experience the world as intensely temporal. It is a book that strikes at the heart of modern physics. It casts doubt on Einstein's greatest contribution, the spacetime continuum, but also points to the solution of one of the great paradoxes of modern science, the chasm between classical and quantum physics. Indeed, Barbour argues that the holy grail of physicists--the unification of Einstein's general relativity with quantum mechanics--may well spell the end of time. Barbour writes with remarkable clarity as he ranges from the ancient philosophers Heraclitus and Parmenides, through the giants of science Galileo, Newton, and Einstein, to the work of the contemporary physicists John Wheeler, Roger Penrose, and Steven Hawking. Along the way he treats us to enticing glimpses of some of the mysteries of the universe, and presents intriguing ideas about multiple worlds, time travel, immortality, and, above all, the illusion of motion. The End of Time is a vibrantly written and revolutionary book. It turns our understanding of reality inside-out.

@article{doi105860choice380352,
    title = "The end of time: the next revolution in physics",
    year = "2000",
    journal = "Choice Reviews Online",
    abstract = "The End of Time: The Next Revolution in Physics by Julian Barbour Richard Feynman once quipped that is what happens when nothing else does. But Julian Barbour disagrees: if nothing happened, if nothing changed, then time would stop. For time is nothing but change. It is change that we perceive occurring all around us, not time. Put simply, time does not exist. In this highly provocative volume, Barbour presents the basic evidence for a timeless universe, and shows why we still experience the world as intensely temporal. It is a book that strikes at the heart of modern physics. It casts doubt on Einstein's greatest contribution, the spacetime continuum, but also points to the solution of one of the great paradoxes of modern science, the chasm between classical and quantum physics. Indeed, Barbour argues that the holy grail of physicists--the unification of Einstein's general relativity with quantum mechanics--may well spell the end of time. Barbour writes with remarkable clarity as he ranges from the ancient philosophers Heraclitus and Parmenides, through the giants of science Galileo, Newton, and Einstein, to the work of the contemporary physicists John Wheeler, Roger Penrose, and Steven Hawking. Along the way he treats us to enticing glimpses of some of the mysteries of the universe, and presents intriguing ideas about multiple worlds, time travel, immortality, and, above all, the illusion of motion. The End of Time is a vibrantly written and revolutionary book. It turns our understanding of reality inside-out.",
    url = "https://doi.org/10.5860/choice.38-0352",
    doi = "10.5860/choice.38-0352",
    openalex = "W1587936857"
}

Some principles underpinning the running of the Universe are discussed. The most important, the machine principle, states that the Universe is a fully autonomous, self-organizing and self-testing quantum automaton. Continuous space and time, consciousness and the semi-classical observers of quantum mechanics are all emergent phenomena not operating at the fundamental level of the machine Universe. Quantum processes define the present, the interface between the future and the past, giving a time ordering to the running of the Universe which is non-integrable except on emergent scales. A diagrammatic approach is used to discuss the quantum topology of the EPR paradox, particle decays and scattering processes. A toy model of a self-referential universe is given.

@misc{doi1048550arxivquantph0105013,
    author = "Jaroszkiewicz, George",
    title = "The running of the Universe and the quantum structure of time",
    year = "2001",
    booktitle = "arXiv (Cornell University)",
    abstract = "Some principles underpinning the running of the Universe are discussed. The most important, the machine principle, states that the Universe is a fully autonomous, self-organizing and self-testing quantum automaton. Continuous space and time, consciousness and the semi-classical observers of quantum mechanics are all emergent phenomena not operating at the fundamental level of the machine Universe. Quantum processes define the present, the interface between the future and the past, giving a time ordering to the running of the Universe which is non-integrable except on emergent scales. A diagrammatic approach is used to discuss the quantum topology of the EPR paradox, particle decays and scattering processes. A toy model of a self-referential universe is given.",
    url = "https://doi.org/10.48550/arxiv.quant-ph/0105013",
    doi = "10.48550/arxiv.quant-ph/0105013",
    openalex = "W1761165074",
    references = "doi101007978146154285813"
}

Causality in classical field theories must be inserted by hand by choosing the retarded solution. It is shown how apparent contradictions in the Coulomb gauge can be resolved and that a causal Coulomb field exists despite the appearance to the contrary. Similarly, it is shown how Newtonian gravitation leads from action-at-a-distance to a causal field when a first-order correction for space–time curvature is applied.

@article{doi10111911435345,
    author = "Rohrlich, F.",
    title = "Causality, the Coulomb field, and Newton’s law of gravitation",
    year = "2002",
    journal = "American Journal of Physics",
    abstract = "Causality in classical field theories must be inserted by hand by choosing the retarded solution. It is shown how apparent contradictions in the Coulomb gauge can be resolved and that a causal Coulomb field exists despite the appearance to the contrary. Similarly, it is shown how Newtonian gravitation leads from action-at-a-distance to a causal field when a first-order correction for space–time curvature is applied.",
    url = "https://doi.org/10.1119/1.1435345",
    doi = "10.1119/1.1435345",
    openalex = "W1985441238",
    references = "doi101016s1355219899000301"
}

@article{doi101016jshpsb200402005,
    author = "Rovelli, Carlo",
    title = "Comment on: “Causality and the arrow of classical time”, by Fritz Rohrlich",
    year = "2004",
    journal = "Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern Physics",
    url = "https://doi.org/10.1016/j.shpsb.2004.02.005",
    doi = "10.1016/j.shpsb.2004.02.005",
    openalex = "W1971928963",
    references = "doi101016s1355219899000301"
}

From information theory and thermodynamic considerations a universal bound on the relaxation time $\ensuremath{\tau}$ of a perturbed system is inferred, $\ensuremath{\tau}\ensuremath{\ge}\ensuremath{\hbar}/\ensuremath{\pi}T$, where $T$ is the system's temperature. We show that black holes comply with the bound; in fact they may actually saturate it. Thus, when judged by their relaxation properties, black holes are the most extreme objects in nature, having the maximum relaxation rate which is allowed by quantum theory.

@article{doi101103physrevd75064013,
    author = "Hod, Shahar",
    title = "Universal bound on dynamical relaxation times and black-hole quasinormal ringing",
    year = "2007",
    journal = "Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D, Particles, fields, gravitation, and cosmology",
    abstract = "From information theory and thermodynamic considerations a universal bound on the relaxation time $\ensuremath{\tau}$ of a perturbed system is inferred, $\ensuremath{\tau}\ensuremath{\ge}\ensuremath{\hbar}/\ensuremath{\pi}T$, where $T$ is the system's temperature. We show that black holes comply with the bound; in fact they may actually saturate it. Thus, when judged by their relaxation properties, black holes are the most extreme objects in nature, having the maximum relaxation rate which is allowed by quantum theory.",
    url = "https://doi.org/10.1103/physrevd.75.064013",
    doi = "10.1103/physrevd.75.064013",
    openalex = "W2116008920",
    references = "doi10108000107510310001632523"
}

Granger causality is a statistical notion of causal influence based on prediction via vector autoregression. Developed originally in the field of econometrics, it has since found application in a broader arena, particularly in neuroscience. More recently transfer entropy, an information-theoretic measure of time-directed information transfer between jointly dependent processes, has gained traction in a similarly wide field. While it has been recognized that the two concepts must be related, the exact relationship has until now not been formally described. Here we show that for Gaussian variables, Granger causality and transfer entropy are entirely equivalent, thus bridging autoregressive and information-theoretic approaches to data-driven causal inference.

@article{doi101103physrevlett103238701,
    author = "Barnett, Lionel and Barrett, Adam B. and Seth, Anil K.",
    title = "Granger Causality and Transfer Entropy Are Equivalent for Gaussian Variables",
    year = "2009",
    journal = "Physical Review Letters",
    abstract = "Granger causality is a statistical notion of causal influence based on prediction via vector autoregression. Developed originally in the field of econometrics, it has since found application in a broader arena, particularly in neuroscience. More recently transfer entropy, an information-theoretic measure of time-directed information transfer between jointly dependent processes, has gained traction in a similarly wide field. While it has been recognized that the two concepts must be related, the exact relationship has until now not been formally described. Here we show that for Gaussian variables, Granger causality and transfer entropy are entirely equivalent, thus bridging autoregressive and information-theoretic approaches to data-driven causal inference.",
    url = "https://doi.org/10.1103/physrevlett.103.238701",
    doi = "10.1103/physrevlett.103.238701",
    openalex = "W2079656335"
}

@incollection{vandrongelen2010causality,
    author = "van Drongelen, Wim",
    title = "Causality",
    year = "2010",
    booktitle = "Signal Processing for Neuroscientists",
    url = "https://doi.org/10.1016/b978-0-12-384915-1.00007-3",
    doi = "10.1016/b978-0-12-384915-1.00007-3",
    pages = "159-175"
}

@incollection{crossref2011thinking,
    title = "Thinking with Causality about “Causality”",
    year = "2011",
    booktitle = "Contrary Thinking",
    url = "https://doi.org/10.1093/acprof:osobl/9780199795550.003.0004",
    doi = "10.1093/acprof:osobl/9780199795550.003.0004",
    pages = "46-56"
}

Abstract\nThe notion of thermal time has been introduced as a possible basis for a fully general-relativistic thermodynamics. Here we study this notion in the restricted context of stationary spacetimes. We show that the Tolman-Ehrenfest effect (in a stationary gravitational field, temperature is not constant in space at thermal equilibrium) can be derived very simply by applying the equivalence principle to a key property of thermal time at equilibrium: temperature is the rate of thermal time with respect to proper time? the 'speed of (thermal) time '. Unlike other published derivations of the Tolman-Ehrenfest law, this one is free from any further dynamical assumption, thereby illustrating the physical import of the notion of thermal time.

@article{doi10108802649381287075007,
    author = "Rovelli, Carlo and Smerlak, Matteo",
    title = "Thermal time and Tolman–Ehrenfest effect: ‘temperature as the speed of time’",
    year = "2011",
    journal = "Classical and Quantum Gravity",
    abstract = "Abstract\nThe notion of thermal time has been introduced as a possible basis for a fully general-relativistic thermodynamics. Here we study this notion in the restricted context of stationary spacetimes. We show that the Tolman-Ehrenfest effect (in a stationary gravitational field, temperature is not constant in space at thermal equilibrium) can be derived very simply by applying the equivalence principle to a key property of thermal time at equilibrium: temperature is the rate of thermal time with respect to proper time? the 'speed of (thermal) time '. Unlike other published derivations of the Tolman-Ehrenfest law, this one is free from any further dynamical assumption, thereby illustrating the physical import of the notion of thermal time.",
    url = "https://doi.org/10.1088/0264-9381/28/7/075007",
    doi = "10.1088/0264-9381/28/7/075007",
    openalex = "W2009557756"
}

It is often claimed, or hoped, that some temporal asymmetries are explained by the thermodynamic asymmetry in time. Thermodynamics, the macroscopic physics of pressure, temperature, volume, and so on, describes many temporally asymmetric processes. Heat flows spontaneously from hot objects to cold objects (in closed systems), never the reverse. More generally, systems spontaneously move from non-equilibrium states to equilibrium states, never the reverse. Delving into the foundations of statistical mechanics, this chapter reviews the many open questions in that field as they relate to temporal asymmetry. Taking a stand on many of them, it tackles questions about the nature of probabilities, the role of boundary conditions, and even the nature and scope of statistical mechanics.

@book{doi101093oxfordhb97801992982040030011,
    author = "North, Jill",
    title = "Time in Thermodynamics",
    year = "2011",
    booktitle = "Oxford University Press eBooks",
    abstract = "It is often claimed, or hoped, that some temporal asymmetries are explained by the thermodynamic asymmetry in time. Thermodynamics, the macroscopic physics of pressure, temperature, volume, and so on, describes many temporally asymmetric processes. Heat flows spontaneously from hot objects to cold objects (in closed systems), never the reverse. More generally, systems spontaneously move from non-equilibrium states to equilibrium states, never the reverse. Delving into the foundations of statistical mechanics, this chapter reviews the many open questions in that field as they relate to temporal asymmetry. Taking a stand on many of them, it tackles questions about the nature of probabilities, the role of boundary conditions, and even the nature and scope of statistical mechanics.",
    url = "https://doi.org/10.1093/oxfordhb/9780199298204.003.0011",
    doi = "10.1093/oxfordhb/9780199298204.003.0011",
    openalex = "W1555357518",
    references = "doi101016jshpsb200402005, doi101016s1355219899000301"
}

Time in electromagnetism shares many features with time in other physical theories. But there is one aspect of electromagnetism's relationship with time that has always been controversial, yet has not always attracted the limelight it deserves: the electromagnetic arrow of time. Beginning with a re-analysis of a famous argument between Ritz and Einstein over the origins of the radiation arrow, this chapter frames the debate between modern Einsteinians and neo-Ritzians. It tries to find a clean statement of what the arrow is and then explains how it relates to the cosmological and thermodynamic arrows, representing the most developed and sophisticated attack yet, in either the physics or philosophy literature, on the electromagnetic arrow of time.

@book{doi101093oxfordhb97801992982040030017,
    author = "Earman, John",
    title = "Sharpening the Electromagnetic Arrow(s) of Time",
    year = "2011",
    booktitle = "Oxford University Press eBooks",
    abstract = "Time in electromagnetism shares many features with time in other physical theories. But there is one aspect of electromagnetism's relationship with time that has always been controversial, yet has not always attracted the limelight it deserves: the electromagnetic arrow of time. Beginning with a re-analysis of a famous argument between Ritz and Einstein over the origins of the radiation arrow, this chapter frames the debate between modern Einsteinians and neo-Ritzians. It tries to find a clean statement of what the arrow is and then explains how it relates to the cosmological and thermodynamic arrows, representing the most developed and sophisticated attack yet, in either the physics or philosophy literature, on the electromagnetic arrow of time.",
    url = "https://doi.org/10.1093/oxfordhb/9780199298204.003.0017",
    doi = "10.1093/oxfordhb/9780199298204.003.0017",
    openalex = "W176576763",
    references = "doi101016jshpsb200402005, doi101016s1355219899000301"
}

Identifying causal networks is important for effective policy and management recommendations on climate, epidemiology, financial regulation, and much else. We introduce a method, based on nonlinear state space reconstruction, that can distinguish causality from correlation. It extends to nonseparable weakly connected dynamic systems (cases not covered by the current Granger causality paradigm). The approach is illustrated both by simple models (where, in contrast to the real world, we know the underlying equations/relations and so can check the validity of our method) and by application to real ecological systems, including the controversial sardine-anchovy-temperature problem.

@article{doi101126science1227079,
    author = "Sugihara, George and May, Robert M. and Ye, Hao and Hsieh, Chih‐hao and Deyle, Ethan R. and Fogarty, Michael J. and Munch, Stephan B.",
    title = "Detecting Causality in Complex Ecosystems",
    year = "2012",
    journal = "Science",
    abstract = "Identifying causal networks is important for effective policy and management recommendations on climate, epidemiology, financial regulation, and much else. We introduce a method, based on nonlinear state space reconstruction, that can distinguish causality from correlation. It extends to nonseparable weakly connected dynamic systems (cases not covered by the current Granger causality paradigm). The approach is illustrated both by simple models (where, in contrast to the real world, we know the underlying equations/relations and so can check the validity of our method) and by application to real ecological systems, including the controversial sardine-anchovy-temperature problem.",
    url = "https://doi.org/10.1126/science.1227079",
    doi = "10.1126/science.1227079",
    openalex = "W2083278075",
    references = "doi101007bf01053745, doi101007bfb0091924, doi101038344734a0, doi101103physrevlett45712, doi101103physrevlett85461, doi101111j146800841990mp52002003x, doi1011751520047719980790061apgtwa20co2, doi1023071912017, doi1023071912791, doi1023071913236"
}

If a black hole starts in a pure quantum state and evaporates completely by a unitary process, the von Neumann entropy of the Hawking radiation initially increases and then decreases back to zero when the black hole has disappeared. Here numerical results are given for an approximation to the time dependence of the radiation entropy under an assumption of fast scrambling, for large nonrotating black holes that emit essentially only photons and gravitons. The maximum of the von Neumann entropy then occurs after about 53.81% of the evaporation time, when the black hole has lost about 40.25% of its original Bekenstein-Hawking (BH) entropy (an upper bound for its von Neumann entropy) and then has a BH entropy that equals the entropy in the radiation, which is about 59.75% of the original BH entropy 4 pi M_0^2, or about 7.509 M_0^2 \approx 6.268 x 10^{76}(M_0/M_sun)^2, using my 1976 calculations that the photon and graviton emission process into empty space gives about 1.4847 times the BH entropy loss of the black hole. Results are also given for black holes in initially impure states. If the black hole starts in a maximally mixed state, the von Neumann entropy of the Hawking radiation increases from zero up to a maximum of about 119.51% of the original BH entropy, or about 15.018 M_0^2 \approx 1.254 x 10^{77}(M_0/M_sun)^2, and then decreases back down to 4 pi M_0^2 = 1.049 x 10^{77}(M_0/M_sun)^2.

@article{doi10108814757516201309028,
    author = "Page, Don N.",
    title = "Time dependence of Hawking radiation entropy",
    year = "2013",
    journal = "Journal of Cosmology and Astroparticle Physics",
    abstract = "If a black hole starts in a pure quantum state and evaporates completely by a unitary process, the von Neumann entropy of the Hawking radiation initially increases and then decreases back to zero when the black hole has disappeared. Here numerical results are given for an approximation to the time dependence of the radiation entropy under an assumption of fast scrambling, for large nonrotating black holes that emit essentially only photons and gravitons. The maximum of the von Neumann entropy then occurs after about 53.81\% of the evaporation time, when the black hole has lost about 40.25\% of its original Bekenstein-Hawking (BH) entropy (an upper bound for its von Neumann entropy) and then has a BH entropy that equals the entropy in the radiation, which is about 59.75\% of the original BH entropy 4 pi M\_0^2, or about 7.509 M\_0^2 \approx 6.268 x 10^{76}(M\_0/M\_sun)^2, using my 1976 calculations that the photon and graviton emission process into empty space gives about 1.4847 times the BH entropy loss of the black hole. Results are also given for black holes in initially impure states. If the black hole starts in a maximally mixed state, the von Neumann entropy of the Hawking radiation increases from zero up to a maximum of about 119.51\% of the original BH entropy, or about 15.018 M\_0^2 \approx 1.254 x 10^{77}(M\_0/M\_sun)^2, and then decreases back down to 4 pi M\_0^2 = 1.049 x 10^{77}(M\_0/M\_sun)^2.",
    url = "https://doi.org/10.1088/1475-7516/2013/09/028",
    doi = "10.1088/1475-7516/2013/09/028",
    openalex = "W2110969537",
    references = "doi10108000107510310001632523"
}

We clarify several issues involving the concepts of time-reversal invariance and time asymmetry in classical electrodynamics. Specifically, we consider three questions: (I) If electrodynamics is time-reversal invariant, why are the radiative processes that occur in nature time asymmetric? (II) Why doesn't the time-reversal invariance of electrodynamics contradict the fact that a charged particle in motion feels a radiation damping force? (III) Why don't the advanced solutions to Maxwell's equations occur in nature—is there some principle that forbids them? We argue that these questions are not specific to electrodynamics, but also arise in many other systems in which waves are radiated, and answer them in the context of a simple model describing a coupled particle-field system in (1+1) dimensions.

@article{doi10111914807756,
    author = "Boozer, A. D.",
    title = "Time-reversal invariance and time asymmetry in classical electrodynamics",
    year = "2013",
    journal = "American Journal of Physics",
    abstract = "We clarify several issues involving the concepts of time-reversal invariance and time asymmetry in classical electrodynamics. Specifically, we consider three questions: (I) If electrodynamics is time-reversal invariant, why are the radiative processes that occur in nature time asymmetric? (II) Why doesn't the time-reversal invariance of electrodynamics contradict the fact that a charged particle in motion feels a radiation damping force? (III) Why don't the advanced solutions to Maxwell's equations occur in nature—is there some principle that forbids them? We argue that these questions are not specific to electrodynamics, but also arise in many other systems in which waves are radiated, and answer them in the context of a simple model describing a coupled particle-field system in (1+1) dimensions.",
    url = "https://doi.org/10.1119/1.4807756",
    doi = "10.1119/1.4807756",
    openalex = "W2031362412",
    references = "doi101016jshpsb200402005"
}

Could time be discrete on some unimaginably small scale? Exploring the idea in depth, this unique introduction to discrete time mechanics systematically builds the theory up from scratch, beginning with the historical, physical and mathematical background to the chronon hypothesis. Covering classical and quantum discrete time mechanics, this book presents all the tools needed to formulate and develop applications of discrete time mechanics in a number of areas, including spreadsheet mechanics, classical and quantum register mechanics, and classical and quantum mechanics and field theories. A consistent emphasis on contextuality and the observer-system relationship is maintained throughout.

@book{doi101017cbo9781139525381,
    author = "Jaroszkiewicz, George",
    title = "Principles of Discrete Time Mechanics",
    year = "2014",
    booktitle = "Cambridge University Press eBooks",
    abstract = "Could time be discrete on some unimaginably small scale? Exploring the idea in depth, this unique introduction to discrete time mechanics systematically builds the theory up from scratch, beginning with the historical, physical and mathematical background to the chronon hypothesis. Covering classical and quantum discrete time mechanics, this book presents all the tools needed to formulate and develop applications of discrete time mechanics in a number of areas, including spreadsheet mechanics, classical and quantum register mechanics, and classical and quantum mechanics and field theories. A consistent emphasis on contextuality and the observer-system relationship is maintained throughout.",
    url = "https://doi.org/10.1017/cbo9781139525381",
    doi = "10.1017/cbo9781139525381",
    openalex = "W578618433",
    references = "doi101007978146154285813, doi101017cbo9780511565748, openalexw3125786388"
}

The basic microsopic physical laws are time reversible. In contrast, the second law of thermodynamics, which is a macroscopic physical representation of the world, is able to describe irreversible processes in an isolated system through the change of entropy ΔS > 0. It is the attempt of the present manuscript to bridge the microscopic physical world with its macrosocpic one with an alternative approach than the statistical mechanics theory of Gibbs and Boltzmann. It is proposed that time is discrete with constant step size. Its consequence is the presence of time irreversibility at the microscopic level if the present force is of complex nature (F (r) ≠ const). In order to compare this discrete time irreversible mechamics (for simplicity a “classical”, single particle in a one dimensional space is selected) with its classical Newton analog, time reversibility is reintroduced by scaling the time steps for any given time step n by the variable sn leading to the Nosé-Hoover Lagrangian. The corresponding Nosé-Hoover Hamiltonian comprises a term Ndf kB T ln sn (kB the Boltzmann constant, T the temperature, and Ndf the number of degrees of freedom) which is defined as the microscopic entropySn at time point n multiplied by T. Upon ensemble averaging this microscopic entropy Sn in equilibrium for a system which does not have fast changing forces approximates its macroscopic counterpart known from thermodynamics. The presented derivation with the resulting analogy between the ensemble averaged microscopic entropy and its thermodynamic analog suggests that the original description of the entropy by Boltzmann and Gibbs is just an ensemble averaging of the time scaling variable sn which is in equilibrium close to 1, but that the entropy term itself has its root not in statistical mechanics but rather in the discreteness of time.

@article{doi103929ethzb000087164,
    author = "Riek, Roland",
    title = "A Derivation of a Microscopic Entropy and Time Irreversibility From the Discreteness of Time",
    year = "2014",
    journal = "Repository for Publications and Research Data (ETH Zurich)",
    abstract = "The basic microsopic physical laws are time reversible. In contrast, the second law of thermodynamics, which is a macroscopic physical representation of the world, is able to describe irreversible processes in an isolated system through the change of entropy ΔS > 0. It is the attempt of the present manuscript to bridge the microscopic physical world with its macrosocpic one with an alternative approach than the statistical mechanics theory of Gibbs and Boltzmann. It is proposed that time is discrete with constant step size. Its consequence is the presence of time irreversibility at the microscopic level if the present force is of complex nature (F (r) ≠ const). In order to compare this discrete time irreversible mechamics (for simplicity a “classical”, single particle in a one dimensional space is selected) with its classical Newton analog, time reversibility is reintroduced by scaling the time steps for any given time step n by the variable sn leading to the Nosé-Hoover Lagrangian. The corresponding Nosé-Hoover Hamiltonian comprises a term Ndf kB T ln sn (kB the Boltzmann constant, T the temperature, and Ndf the number of degrees of freedom) which is defined as the microscopic entropySn at time point n multiplied by T. Upon ensemble averaging this microscopic entropy Sn in equilibrium for a system which does not have fast changing forces approximates its macroscopic counterpart known from thermodynamics. The presented derivation with the resulting analogy between the ensemble averaged microscopic entropy and its thermodynamic analog suggests that the original description of the entropy by Boltzmann and Gibbs is just an ensemble averaging of the time scaling variable sn which is in equilibrium close to 1, but that the entropy term itself has its root not in statistical mechanics but rather in the discreteness of time.",
    url = "https://doi.org/10.3929/ethz-b-000087164",
    doi = "10.3929/ethz-b-000087164",
    openalex = "W3103854884",
    references = "openalexw3125786388"
}

Different aspects of the causality concept are discussed, inclusive of the fact that no widely accepted definition of causality exists. Still the concept has been essential to the development of science. Further, aspects of how to establish causal relationships are discussed, including causality and statistics, empirical versus mechanistic causal models, agreement of causal relationships within acceptable margins of error when statistical data are part of establishing such relationships, and Karl Popper's concept of falsifiability. A few concepts and tools related to causality in quality and reliability are briefly mentioned. Newer developments in causality and statistics are mentioned and references to important literature on these developments are given, in addition to references to more established knowledge.

@misc{evandt2014causality,
    author = "Evandt, Øystein",
    title = "Causality",
    year = "2014",
    booktitle = "Wiley StatsRef: Statistics Reference Online",
    abstract = "Different aspects of the causality concept are discussed, inclusive of the fact that no widely accepted definition of causality exists. Still the concept has been essential to the development of science. Further, aspects of how to establish causal relationships are discussed, including causality and statistics, empirical versus mechanistic causal models, agreement of causal relationships within acceptable margins of error when statistical data are part of establishing such relationships, and Karl Popper's concept of falsifiability. A few concepts and tools related to causality in quality and reliability are briefly mentioned. Newer developments in causality and statistics are mentioned and references to important literature on these developments are given, in addition to references to more established knowledge.",
    url = "https://doi.org/10.1002/9781118445112.stat03920",
    doi = "10.1002/9781118445112.stat03920"
}

@incollection{crossref2015perceptual,
    title = "Perceptual Causality and Narrative Causality",
    year = "2015",
    booktitle = "Stories, Meaning, and Experience",
    url = "https://doi.org/10.4324/9781315880488-2",
    doi = "10.4324/9781315880488-2",
    pages = "13-50"
}

We consider higher derivative corrections to the graviton three-point coupling within a weakly coupled theory of gravity. Lorentz invariance allows further structures beyond the one present in the Einstein theory. We argue that these are constrained by causality. We devise a thought experiment involving a high energy scattering process which leads to causality violation if the graviton three-point vertex contains the additional structures. This violation cannot be fixed by adding conventional particles with spins J ≤ 2. But, it can be fixed by adding an infinite tower of extra massive particles with higher spins, J > 2. In AdS theories this implies a constraint on the conformal anomaly coefficients $$ \left|\frac{a-c}{c}\right|\lesssim \frac{1}{\Delta_{\mathrm{gap}}^2} $$ n terms of Δgap, the dimension of the lightest single trace operator with spin J > 2. For inflation, or de Sitter-like solutions, it indicates the existence of massive higher spin particles if the gravity wave non-gaussianity deviates significantly from the one computed in the Einstein theory.

@article{doi101007jhep022016020,
    author = "Camanho, Xián O. and Edelstein, José D. and Maldacena, Juan and Zhiboedov, Alexander",
    title = "Causality constraints on corrections to the graviton three-point coupling",
    year = "2016",
    journal = "Journal of High Energy Physics",
    abstract = "We consider higher derivative corrections to the graviton three-point coupling within a weakly coupled theory of gravity. Lorentz invariance allows further structures beyond the one present in the Einstein theory. We argue that these are constrained by causality. We devise a thought experiment involving a high energy scattering process which leads to causality violation if the graviton three-point vertex contains the additional structures. This violation cannot be fixed by adding conventional particles with spins J ≤ 2. But, it can be fixed by adding an infinite tower of extra massive particles with higher spins, J > 2. In AdS theories this implies a constraint on the conformal anomaly coefficients $$ \left|\frac{a-c}{c}\right|\lesssim \frac{1}{\Delta\_{\mathrm{gap}}^2} $$ n terms of Δgap, the dimension of the lightest single trace operator with spin J > 2. For inflation, or de Sitter-like solutions, it indicates the existence of massive higher spin particles if the gravity wave non-gaussianity deviates significantly from the one computed in the Einstein theory.",
    url = "https://doi.org/10.1007/jhep02(2016)020",
    doi = "10.1007/jhep02(2016)020",
    openalex = "W1752223565"
}

We consider higher derivative corrections to the graviton three-point coupling within a weakly coupled theory of gravity. Lorentz invariance allows further structures beyond the one present in the Einstein theory. We argue that these are constrained by causality. We devise a thought experiment involving a high energy scattering process which leads to causality violation if the graviton three-point vertex contains the additional structures. This violation cannot be fixed by adding conventional particles with spins $J \\leq 2$. But, it can be fixed by adding an infinite tower of extra massive particles with higher spins, $J > 2$. In AdS theories this implies a constraint on the conformal anomaly coefficients $\\left|{a - c \\over c} \\right| \\lesssim {1 \\over \\Delta_{gap}^2}$ in terms of $\\Delta_{gap}$, the dimension of the lightest single particle operator with spin $J > 2$. For inflation, or de Sitter-like solutions, it indicates the existence of massive higher spin particles if the gravity wave non-gaussianity deviates significantly from the one computed in the Einstein theory.

@article{doi101007jhep0228201629020,
    author = "Camanho, Xián O.",
    title = "Causality Constraints on Corrections to the Graviton Three-Point Coupling",
    year = "2016",
    journal = "MPG.PuRe (Max Planck Society)",
    abstract = "We consider higher derivative corrections to the graviton three-point coupling within a weakly coupled theory of gravity. Lorentz invariance allows further structures beyond the one present in the Einstein theory. We argue that these are constrained by causality. We devise a thought experiment involving a high energy scattering process which leads to causality violation if the graviton three-point vertex contains the additional structures. This violation cannot be fixed by adding conventional particles with spins $J \\leq 2$. But, it can be fixed by adding an infinite tower of extra massive particles with higher spins, $J > 2$. In AdS theories this implies a constraint on the conformal anomaly coefficients $\\left|{a - c \\over c} \\right| \\lesssim {1 \\over \\Delta\_{gap}^2}$ in terms of $\\Delta\_{gap}$, the dimension of the lightest single particle operator with spin $J > 2$. For inflation, or de Sitter-like solutions, it indicates the existence of massive higher spin particles if the gravity wave non-gaussianity deviates significantly from the one computed in the Einstein theory.",
    url = "https://doi.org/10.1007/jhep02\%282016\%29020",
    doi = "10.1007/jhep02\%282016\%29020",
    openalex = "W3101868850"
}

We study the dynamics of photonic quantum circuits consisting of nodes coupled by quantum channels. We are interested in the regime where the time delay in communication between the nodes is significant. This includes the problem of quantum feedback, where a quantum signal is fed back on a system with a time delay. We develop a matrix product state approach to solve the quantum stochastic Schrödinger equation with time delays, which accounts in an efficient way for the entanglement of nodes with the stream of emitted photons in the waveguide, and thus the non-Markovian character of the dynamics. We illustrate this approach with two paradigmatic quantum optical examples: two coherently driven distant atoms coupled to a photonic waveguide with a time delay, and a driven atom coupled to its own output field with a time delay as an instance of a quantum feedback problem.

@article{doi101103physrevlett116093601,
    author = "Pichler, Hannes and Zoller, P.",
    title = "Photonic Circuits with Time Delays and Quantum Feedback",
    year = "2016",
    journal = "Physical Review Letters",
    abstract = "We study the dynamics of photonic quantum circuits consisting of nodes coupled by quantum channels. We are interested in the regime where the time delay in communication between the nodes is significant. This includes the problem of quantum feedback, where a quantum signal is fed back on a system with a time delay. We develop a matrix product state approach to solve the quantum stochastic Schrödinger equation with time delays, which accounts in an efficient way for the entanglement of nodes with the stream of emitted photons in the waveguide, and thus the non-Markovian character of the dynamics. We illustrate this approach with two paradigmatic quantum optical examples: two coherently driven distant atoms coupled to a photonic waveguide with a time delay, and a driven atom coupled to its own output field with a time delay as an instance of a quantum feedback problem.",
    url = "https://doi.org/10.1103/physrevlett.116.093601",
    doi = "10.1103/physrevlett.116.093601",
    openalex = "W2283341665",
    references = "doi101103physreva101096"
}

@incollection{crossref2017causality,
    title = "Causality",
    year = "2017",
    booktitle = "Encyclopedia of Educational Philosophy and Theory",
    url = "https://doi.org/10.1007/978-981-287-588-4\_100075",
    doi = "10.1007/978-981-287-588-4\_100075",
    pages = "102-102"
}

@incollection{silva2017causality,
    author = "Silva, Ricardo",
    title = "Causality",
    year = "2017",
    booktitle = "Encyclopedia of Machine Learning and Data Mining",
    url = "https://doi.org/10.1007/978-1-4899-7687-1\_36",
    doi = "10.1007/978-1-4899-7687-1\_36",
    pages = "194-202"
}

Using several methods for detection of causality in time series, we show in a numerical study that coupled chaotic dynamical systems violate the first principle of Granger causality that the cause precedes the effect. While such a violation can be observed in formal applications of time series analysis methods, it cannot occur in nature, due to the relation between entropy production and temporal irreversibility. The obtained knowledge, however, can help to understand the type of causal relations observed in experimental data, namely, it can help to distinguish linear transfer of time-delayed signals from nonlinear interactions. We illustrate these findings in causality detected in experimental time series from the climate system and mammalian cardio-respiratory interactions.

@article{doi10106315019944,
    author = "Paluš, Milan and Krakovská, Anna and Jakubík, Jozef and Chvosteková, Martina",
    title = "Causality, dynamical systems and the arrow of time",
    year = "2018",
    journal = "Chaos An Interdisciplinary Journal of Nonlinear Science",
    abstract = "Using several methods for detection of causality in time series, we show in a numerical study that coupled chaotic dynamical systems violate the first principle of Granger causality that the cause precedes the effect. While such a violation can be observed in formal applications of time series analysis methods, it cannot occur in nature, due to the relation between entropy production and temporal irreversibility. The obtained knowledge, however, can help to understand the type of causal relations observed in experimental data, namely, it can help to distinguish linear transfer of time-delayed signals from nonlinear interactions. We illustrate these findings in causality detected in experimental time series from the climate system and mammalian cardio-respiratory interactions.",
    url = "https://doi.org/10.1063/1.5019944",
    doi = "10.1063/1.5019944",
    openalex = "W2884343086",
    references = "doi101007bfb0091924, doi101016s0370157302001370, doi101016s1053811903002027, doi101070rm1977v032n04abeh001639, doi101103physreva331134, doi101103physrevlett103238701, doi101103physrevlett784193, doi101103physrevlett85461, doi101126science1227079, doi1023071912791"
}

Causality in quantum field theory is defined by the vanishing of field commutators for spacelike separations. However, this does not imply a direction for causal effects. Hidden in our conventions for quantization is a connection to the definition of an arrow of causality, i.e., what is the past and what is the future. If we mix quantization conventions within the same theory, we get a violation of microcausality. In such a theory with mixed conventions, the dominant definition of the arrow of causality is determined by the stable states. In some quantum gravity theories, such as quadratic gravity and possibly asymptotic safety, such a mixed causality condition occurs. We discuss some of the implications.

@article{doi101103physrevlett123171601,
    author = "Donoghue, John F. and Menezes, Gabriel",
    title = "Arrow of Causality and Quantum Gravity",
    year = "2019",
    journal = "Physical Review Letters",
    abstract = "Causality in quantum field theory is defined by the vanishing of field commutators for spacelike separations. However, this does not imply a direction for causal effects. Hidden in our conventions for quantization is a connection to the definition of an arrow of causality, i.e., what is the past and what is the future. If we mix quantization conventions within the same theory, we get a violation of microcausality. In such a theory with mixed conventions, the dominant definition of the arrow of causality is determined by the stable states. In some quantum gravity theories, such as quadratic gravity and possibly asymptotic safety, such a mixed causality condition occurs. We discuss some of the implications.",
    url = "https://doi.org/10.1103/physrevlett.123.171601",
    doi = "10.1103/physrevlett.123.171601",
    openalex = "W2968394523"
}

@article{doi101016jppnp2020103812,
    author = "Donoghue, John F. and Menezes, Gabriel",
    title = "Quantum causality and the arrows of time and thermodynamics",
    year = "2020",
    journal = "Progress in Particle and Nuclear Physics",
    url = "https://doi.org/10.1016/j.ppnp.2020.103812",
    doi = "10.1016/j.ppnp.2020.103812",
    openalex = "W3047557861",
    references = "doi1010079783540680017, doi101007jhep062014080, doi1010160550321369900984, doi10106311703676, doi101103physrev951612, doi101103physreva101096, doi101103physrevd16953, doi101103revmodphys487, doi10111911973770, openalexw2798647535"
}

Abstract The concept of causal interactions between components is an integral part of hydrology and Earth system sciences. Modelers, decision makers, scientists, and other water resources stakeholders all utilize some notion of cause‐and‐effect to understand processes, make decisions, and infer how systems react to change. However, there are different perspectives on the meaning of causality in complex systems and, further, different frameworks and methodologies with which to detect causal interactions. We propose here that information theory (IT) provides a compelling framework for the detection of causality and discuss approaches for several levels of analyses that capture interactions that range from pairwise to multivariate in nature. We illustrate these types of analyses with an example based on weather station time series variables, in which variables may interact pairwise or jointly and on short to long time scales. In general, many unsolved or even unanticipated questions in the hydrologic sciences could benefit from this perspective.

@article{doi1010292019wr024940,
    author = "Goodwell, Allison E. and Jiang, Peishi and Ruddell, Benjamin L. and Kumar, Praveen",
    title = "Debates—Does Information Theory Provide a New Paradigm for Earth Science? Causality, Interaction, and Feedback",
    year = "2020",
    journal = "Water Resources Research",
    abstract = "Abstract The concept of causal interactions between components is an integral part of hydrology and Earth system sciences. Modelers, decision makers, scientists, and other water resources stakeholders all utilize some notion of cause‐and‐effect to understand processes, make decisions, and infer how systems react to change. However, there are different perspectives on the meaning of causality in complex systems and, further, different frameworks and methodologies with which to detect causal interactions. We propose here that information theory (IT) provides a compelling framework for the detection of causality and discuss approaches for several levels of analyses that capture interactions that range from pairwise to multivariate in nature. We illustrate these types of analyses with an example based on weather station time series variables, in which variables may interact pairwise or jointly and on short to long time scales. In general, many unsolved or even unanticipated questions in the hydrologic sciences could benefit from this perspective.",
    url = "https://doi.org/10.1029/2019wr024940",
    doi = "10.1029/2019wr024940",
    openalex = "W3009145004",
    references = "doi10106315019944"
}

Detecting causality from observational data is a challenging problem. Here, we propose a machine learning based causality approach, Reservoir Computing Causality (RCC), in order to systematically identify causal relationships between variables. We demonstrate that RCC is able to identify the causal direction, coupling delay, and causal chain relations from time series. Compared to a well-known phase space reconstruction based causality method, Extended Convergent Cross Mapping, RCC does not require the estimation of the embedding dimension and delay time. Moreover, RCC has three additional advantages: (i) robustness to noisy time series; (ii) computational efficiency; and (iii) seamless causal inference from high-dimensional data. We also illustrate the power of RCC in identifying remote causal interactions of high-dimensional systems and demonstrate its usability on a real-world example using atmospheric circulation data. Our results suggest that RCC can accurately detect causal relationships in complex systems.

@article{doi10106350007670,
    author = "Huang, Yu and Fu, Zuntao and Franzke, Christian L. E.",
    title = "Detecting causality from time series in a machine learning framework",
    year = "2020",
    journal = "Chaos An Interdisciplinary Journal of Nonlinear Science",
    abstract = "Detecting causality from observational data is a challenging problem. Here, we propose a machine learning based causality approach, Reservoir Computing Causality (RCC), in order to systematically identify causal relationships between variables. We demonstrate that RCC is able to identify the causal direction, coupling delay, and causal chain relations from time series. Compared to a well-known phase space reconstruction based causality method, Extended Convergent Cross Mapping, RCC does not require the estimation of the embedding dimension and delay time. Moreover, RCC has three additional advantages: (i) robustness to noisy time series; (ii) computational efficiency; and (iii) seamless causal inference from high-dimensional data. We also illustrate the power of RCC in identifying remote causal interactions of high-dimensional systems and demonstrate its usability on a real-world example using atmospheric circulation data. Our results suggest that RCC can accurately detect causal relationships in complex systems.",
    url = "https://doi.org/10.1063/5.0007670",
    doi = "10.1063/5.0007670",
    openalex = "W3033702084",
    references = "doi10106315019944"
}

State-space reconstruction is essential to analyze the dynamics and internal interactions of complex systems. However, it is difficult to estimate high-dimensional conditional mutual information and select the optimal time delays in most existing nonuniform state-space reconstruction methods. Therefore, we propose a nonuniform embedding method framed in information theory for state-space reconstruction. We use a low-dimensional approximation of conditional mutual information criterion for time delay selection, which is effectively solved by the particle swarm optimization algorithm. The obtained embedded vector has relatively strong independence and low redundancy, which better characterizes multivariable complex systems and detects coupling within complex systems. In addition, the proposed nonuniform embedding method exhibits good performance in coupling detection of linear stochastic, nonlinear stochastic, chaotic systems. In the actual application, the importance of small airports that cause delay propagation has been demonstrated by constructing the delay propagation network.

@article{doi101103physreve101062113,
    author = "Jia, Ziyu and Lin, Youfang and Liu, Yunxiao and Jiao, Zehui and Wang, Jing",
    title = "Refined nonuniform embedding for coupling detection in multivariate time series",
    year = "2020",
    journal = "Physical review. E",
    abstract = "State-space reconstruction is essential to analyze the dynamics and internal interactions of complex systems. However, it is difficult to estimate high-dimensional conditional mutual information and select the optimal time delays in most existing nonuniform state-space reconstruction methods. Therefore, we propose a nonuniform embedding method framed in information theory for state-space reconstruction. We use a low-dimensional approximation of conditional mutual information criterion for time delay selection, which is effectively solved by the particle swarm optimization algorithm. The obtained embedded vector has relatively strong independence and low redundancy, which better characterizes multivariable complex systems and detects coupling within complex systems. In addition, the proposed nonuniform embedding method exhibits good performance in coupling detection of linear stochastic, nonlinear stochastic, chaotic systems. In the actual application, the importance of small airports that cause delay propagation has been demonstrated by constructing the delay propagation network.",
    url = "https://doi.org/10.1103/physreve.101.062113",
    doi = "10.1103/physreve.101.062113",
    openalex = "W3033188742",
    references = "doi10106315019944"
}

Contemporary physics, with two Einstein's theories (called "relativity" what can be interpreted erroneously) and with Heisenberg's principle of indeterminacy (better: "lack of epistemic determinism") are frequently interpreted as a removal of the causality from physics. We argue that this is wrong. There are no indications in physics, either classical or quantum, that physical laws are indeterministic, on the ontological level. On the other hand, both classical and quantum physics are, practically, indeterministic on the epistemic level: there are no means for us to predict the detailed future of the world. Additionally, essentially all physical principles, including the arrow of time and the conservation of energy could be, hypothetically, violated (with some exceptions in the world of heavier quarks, and probably, the cosmological arrow of time). However, in contrast to Hume's skepticism, we have no experimental evidence that the causality can be removed or even "hung on" in any case. The text contains some didactical-like issues, as well.

@article{doi1029202philcosm242,
    author = "Karwasz, Grzegorz P.",
    title = "Between Physics and Metaphysics — on Determinism, Arrow of Time and Causality",
    year = "2020",
    journal = "Philosophy and Cosmology",
    abstract = {Contemporary physics, with two Einstein's theories (called "relativity" what can be interpreted erroneously) and with Heisenberg's principle of indeterminacy (better: "lack of epistemic determinism") are frequently interpreted as a removal of the causality from physics. We argue that this is wrong. There are no indications in physics, either classical or quantum, that physical laws are indeterministic, on the ontological level. On the other hand, both classical and quantum physics are, practically, indeterministic on the epistemic level: there are no means for us to predict the detailed future of the world. Additionally, essentially all physical principles, including the arrow of time and the conservation of energy could be, hypothetically, violated (with some exceptions in the world of heavier quarks, and probably, the cosmological arrow of time). However, in contrast to Hume's skepticism, we have no experimental evidence that the causality can be removed or even "hung on" in any case. The text contains some didactical-like issues, as well.},
    url = "https://doi.org/10.29202/phil-cosm/24/2",
    doi = "10.29202/phil-cosm/24/2",
    openalex = "W3001756571",
    references = "doi10106311580050, doi10108802649381115001, doi101103physrev128919, doi101103physrev4873, doi101103physrev59223, doi101103physrevlett801121, doi10111911356698, doi1043249780203995150, doi105860choice520330, openalexw600996220"
}

@article{donoghue2020quantum,
    author = "Donoghue, John F. and Menezes, Gabriel",
    title = "Quantum causality and the arrows of time and thermodynamics",
    year = "2020",
    journal = "Progress in Particle and Nuclear Physics",
    url = "https://doi.org/10.1016/j.ppnp.2020.103812",
    doi = "10.1016/j.ppnp.2020.103812",
    openalex = "W3047557861",
    pages = "103812",
    volume = "115",
    references = "doi1010079783540680017, doi101007jhep062014080, doi1010160550321369900984, doi10106311703676, doi101103physrev951612, doi101103physreva101096, doi101103physrevd16953, doi101103revmodphys487, doi10111911973770, openalexw2798647535"
}

Abstract In this contribution, we identify two scenarios for the evolutionary branch cut universe. In the first scenario, the universe evolves continuously from the negative complex cosmological time sector, prior to a primordial singularity, to the positive one, circumventing continuously a branch cut, and no primordial singularity occurs in the imaginary sector, only branch points. In the second scenario, the branch cut and branch point disappear after the realization of the imaginary component of the complex time by means of a Wick rotation, which is replaced by the thermal time. In the second scenario, the universe has its origin in the Big Bang, but the model contemplates simultaneously a mirrored parallel evolutionary universe going backwards in the cosmological thermal time negative sector. A quantum formulation based on the Wheeler–DeWitt equation is sketched and preliminary conclusions are drawn.

@article{doi101002asna202113993,
    author = "Vasconcellos, César A. Zen and Hess, Peter O. and Hadjimichef, Dimiter and Bodmann, B. and RAZEIRA, MOISÉS and Volkmer, G. L.",
    title = "Pushing the limits of time beyond the Big Bang singularity: Scenarios for the branch cut universe",
    year = "2021",
    journal = "Astronomische Nachrichten",
    abstract = "Abstract In this contribution, we identify two scenarios for the evolutionary branch cut universe. In the first scenario, the universe evolves continuously from the negative complex cosmological time sector, prior to a primordial singularity, to the positive one, circumventing continuously a branch cut, and no primordial singularity occurs in the imaginary sector, only branch points. In the second scenario, the branch cut and branch point disappear after the realization of the imaginary component of the complex time by means of a Wick rotation, which is replaced by the thermal time. In the second scenario, the universe has its origin in the Big Bang, but the model contemplates simultaneously a mirrored parallel evolutionary universe going backwards in the cosmological thermal time negative sector. A quantum formulation based on the Wheeler–DeWitt equation is sketched and preliminary conclusions are drawn.",
    url = "https://doi.org/10.1002/asna.202113993",
    doi = "10.1002/asna.202113993",
    openalex = "W3174775746",
    references = "doi101002andp19153521505, doi10100797894011198016, doi101007jhep042018147, doi101017cbo9780511755804, doi101049sqj19660063, doi10108802649381287075007, doi101093mnras935325, doi101103physrev1601113, doi101103physrevlett121161102, doi101103physrevlett93211101"
}

Determinism, causality, chance, free will and divine providence form a class of interlaced problems lying in three domains: philosophy, theology, and physics. Recent article by Dariusz Łukasiewicz in Roczniki Filozoficzne (no. 3, 2020) is a great example. Classical physics, that of Newton and Laplace, may lead to deism: God created the world, but then it goes like a mechanical clock. Quantum mechanics brought some “hope” for a rather naïve theology: God acts in gaps between quanta of indetermination. Obviously, any strict determinism jeopardizes the existence of free will. Yes, but only if human mind follows the laws of physics and only if nothing exists outside the physical limits of space and time. We argue that human action lies in-between two worlds: “earth” and “heavens” using the language of Genesis. In that immaterial world, outside time and space constraints, there is no place for the chain of deterministic events. We discuss, in turn, that the principle of causality, a superior law even in physics, reigns also in the non-material world. Though, determinism in the material universe and causality in both worlds seem to be sufficient conditions, to eliminate “chaotic”, or probabilistic causes from human (and divine) action.

@article{doi1018290rf216941,
    author = "Karwasz, Grzegorz P.",
    title = "On Determinism, Causality, and Free Will: Contribution from Physics",
    year = "2021",
    journal = "Roczniki Filozoficzne",
    abstract = "Determinism, causality, chance, free will and divine providence form a class of interlaced problems lying in three domains: philosophy, theology, and physics. Recent article by Dariusz Łukasiewicz in Roczniki Filozoficzne (no. 3, 2020) is a great example. Classical physics, that of Newton and Laplace, may lead to deism: God created the world, but then it goes like a mechanical clock. Quantum mechanics brought some “hope” for a rather naïve theology: God acts in gaps between quanta of indetermination. Obviously, any strict determinism jeopardizes the existence of free will. Yes, but only if human mind follows the laws of physics and only if nothing exists outside the physical limits of space and time. We argue that human action lies in-between two worlds: “earth” and “heavens” using the language of Genesis. In that immaterial world, outside time and space constraints, there is no place for the chain of deterministic events. We discuss, in turn, that the principle of causality, a superior law even in physics, reigns also in the non-material world. Though, determinism in the material universe and causality in both worlds seem to be sufficient conditions, to eliminate “chaotic”, or probabilistic causes from human (and divine) action.",
    url = "https://doi.org/10.18290/rf21694-1",
    doi = "10.18290/rf21694-1",
    openalex = "W4200600165",
    references = "doi1029202philcosm242"
}

An information-theoretic approach for detecting causality and information transfer is used to identify interactions of solar activity and interplanetary medium conditions with the Earth's magnetosphere-ionosphere systems. A causal information transfer from the solar wind parameters to geomagnetic indices is detected. The vertical component of the interplanetary magnetic field (Bz) influences the auroral electrojet (AE) index with an information transfer delay of 10 min and the geomagnetic disturbances at mid-latitudes measured by the symmetric field in the H component (SYM-H) index with a delay of about 30 min. Using a properly conditioned causality measure, no causal link between AE and SYM-H, or between magnetospheric substorms and magnetic storms can be detected. The observed causal relations can be described as linear time-delayed information transfer.

@article{doi103390e23040390,
    author = "Manshour, Pouya and Balasis, Georgios and Consolini, Giuseppe and Papadimitriou, Constantinos and Paluš, Milan",
    title = "Causality and Information Transfer Between the Solar Wind and the Magnetosphere–Ionosphere System",
    year = "2021",
    journal = "Entropy",
    abstract = "An information-theoretic approach for detecting causality and information transfer is used to identify interactions of solar activity and interplanetary medium conditions with the Earth's magnetosphere-ionosphere systems. A causal information transfer from the solar wind parameters to geomagnetic indices is detected. The vertical component of the interplanetary magnetic field (Bz) influences the auroral electrojet (AE) index with an information transfer delay of 10 min and the geomagnetic disturbances at mid-latitudes measured by the symmetric field in the H component (SYM-H) index with a delay of about 30 min. Using a properly conditioned causality measure, no causal link between AE and SYM-H, or between magnetospheric substorms and magnetic storms can be detected. The observed causal relations can be described as linear time-delayed information transfer.",
    url = "https://doi.org/10.3390/e23040390",
    doi = "10.3390/e23040390",
    openalex = "W3137353578",
    references = "doi10106315019944"
}

Causality analysis is an important problem lying at the heart of science, and is of particular importance in data science and machine learning. An endeavor during the past 16 years viewing causality as a real physical notion so as to formulate it from first principles, however, seems to have gone unnoticed. This study introduces to the community this line of work, with a long-due generalization of the information flow-based bivariate time series causal inference to multivariate series, based on the recent advance in theoretical development. The resulting formula is transparent, and can be implemented as a computationally very efficient algorithm for application. It can be normalized and tested for statistical significance. Different from the previous work along this line where only information flows are estimated, here an algorithm is also implemented to quantify the influence of a unit to itself. While this forms a challenge in some causal inferences, here it comes naturally, and hence the identification of self-loops in a causal graph is fulfilled automatically as the causalities along edges are inferred. To demonstrate the power of the approach, presented here are two applications in extreme situations. The first is a network of multivariate processes buried in heavy noises (with the noise-to-signal ratio exceeding 100), and the second a network with nearly synchronized chaotic oscillators. In both graphs, confounding processes exist. While it seems to be a challenge to reconstruct from given series these causal graphs, an easy application of the algorithm immediately reveals the desideratum. Particularly, the confounding processes have been accurately differentiated. Considering the surge of interest in the community, this study is very timely.

@article{doi103390e23060679,
    author = "Liang, X. San",
    title = "Normalized Multivariate Time Series Causality Analysis and Causal Graph Reconstruction",
    year = "2021",
    journal = "PubMed Central",
    abstract = "Causality analysis is an important problem lying at the heart of science, and is of particular importance in data science and machine learning. An endeavor during the past 16 years viewing causality as a real physical notion so as to formulate it from first principles, however, seems to have gone unnoticed. This study introduces to the community this line of work, with a long-due generalization of the information flow-based bivariate time series causal inference to multivariate series, based on the recent advance in theoretical development. The resulting formula is transparent, and can be implemented as a computationally very efficient algorithm for application. It can be normalized and tested for statistical significance. Different from the previous work along this line where only information flows are estimated, here an algorithm is also implemented to quantify the influence of a unit to itself. While this forms a challenge in some causal inferences, here it comes naturally, and hence the identification of self-loops in a causal graph is fulfilled automatically as the causalities along edges are inferred. To demonstrate the power of the approach, presented here are two applications in extreme situations. The first is a network of multivariate processes buried in heavy noises (with the noise-to-signal ratio exceeding 100), and the second a network with nearly synchronized chaotic oscillators. In both graphs, confounding processes exist. While it seems to be a challenge to reconstruct from given series these causal graphs, an easy application of the algorithm immediately reveals the desideratum. Particularly, the confounding processes have been accurately differentiated. Considering the surge of interest in the community, this study is very timely.",
    url = "https://doi.org/10.3390/e23060679",
    doi = "10.3390/e23060679",
    openalex = "W3165761151",
    references = "doi10106315019944"
}

Abstract We basis our initial analysis of the arrow of time on a relationship between the time evolution operator of quantum system and the time‐independent density operator, which describes the equilibrium state of a many‐particle system at temperature. We highlight through this analysis the identification of the imaginary temporal component of the branch‐cut complex cosmic form factor with the direction in which the time parameter flows globally, or the arrow of time. As a novelty, in this work, we calculate the number of branches in the branch‐cut universe to achieve causality involving the global time of evolution of the universe and the local time of travel of the light around each Hubble horizon. The preliminary result obtained is comparable to 60 e‐folds of contraction in the FLRW cosmic scale factor to overcome causality achieved in the bouncing model.

@article{doi101002asna20220086,
    author = "Bodmann, B. and Vasconcellos, César A. Zen and de Freitas Pacheco, José and Hess, Peter O. and Hadjimichef, Dimiter",
    title = "Causality and the arrow of time in the branch‐cut cosmology",
    year = "2022",
    journal = "Astronomische Nachrichten",
    abstract = "Abstract We basis our initial analysis of the arrow of time on a relationship between the time evolution operator of quantum system and the time‐independent density operator, which describes the equilibrium state of a many‐particle system at temperature. We highlight through this analysis the identification of the imaginary temporal component of the branch‐cut complex cosmic form factor with the direction in which the time parameter flows globally, or the arrow of time. As a novelty, in this work, we calculate the number of branches in the branch‐cut universe to achieve causality involving the global time of evolution of the universe and the local time of travel of the light around each Hubble horizon. The preliminary result obtained is comparable to 60 e‐folds of contraction in the FLRW cosmic scale factor to overcome causality achieved in the bouncing model.",
    url = "https://doi.org/10.1002/asna.20220086",
    doi = "10.1002/asna.20220086",
    openalex = "W4311112706",
    references = "doi101002asna202113993, doi101007bf00670342, doi101016jphysletb201906056, doi10105100046361201833910, doi10108000107510310001632523, doi10108813616382aac482, doi101093mnras1166662, doi101103physrevd322489, doi101103physrevd64123522, doi101103physrevd89023525"
}

Research on concepts and computational methods of causality has a long history, and there are various concepts of causality as well as corresponding computing algorithms based on measured data. Here, by considering causes and effects from a dynamical perspective, we present a unified mathematical framework for the so-called dynamical causality (DC), which can detect causal interactions over time; notably, this framework covers Granger causality, transfer entropy, embedding causality and their conditional versions. Based on this framework, we further propose a causality criterion called embedding entropy (EE) to measure the DC between two variables. Moreover, its conditional version, conditional embedding entropy (cEE), is also derived for detecting conditional/direct causality. The significant advantages of EE and cEE are that they can be employed for solving not only nonlinear causal inference but also the non-separability problem, and they can reduce the scale bias in numerical calculation. The performance and robustness of EE and cEE were demonstrated through numerical simulations, and causal inference on various real-world datasets validated their effectiveness.

@article{doi101098rsif20210766,
    author = "Shi, Jifan and Chen, Luonan and Aihara, Kazuyuki",
    title = "Embedding entropy: a nonlinear measure of dynamical causality",
    year = "2022",
    journal = "Journal of The Royal Society Interface",
    abstract = "Research on concepts and computational methods of causality has a long history, and there are various concepts of causality as well as corresponding computing algorithms based on measured data. Here, by considering causes and effects from a dynamical perspective, we present a unified mathematical framework for the so-called dynamical causality (DC), which can detect causal interactions over time; notably, this framework covers Granger causality, transfer entropy, embedding causality and their conditional versions. Based on this framework, we further propose a causality criterion called embedding entropy (EE) to measure the DC between two variables. Moreover, its conditional version, conditional embedding entropy (cEE), is also derived for detecting conditional/direct causality. The significant advantages of EE and cEE are that they can be employed for solving not only nonlinear causal inference but also the non-separability problem, and they can reduce the scale bias in numerical calculation. The performance and robustness of EE and cEE were demonstrated through numerical simulations, and causal inference on various real-world datasets validated their effectiveness.",
    url = "https://doi.org/10.1098/rsif.2021.0766",
    doi = "10.1098/rsif.2021.0766",
    openalex = "W4220791818",
    references = "doi10106315019944"
}

The directional dependencies of different climate indices are explored using the Liang-Kleeman information flow in order to disentangle the influence of certain regions over the globe on the development of low-frequency variability of others. Seven key indices (the sea-surface temperature in El-Niño 3.4 region, the Atlantic Multidecadal Oscillation, the North Atlantic Oscillation, the North Pacific America pattern, the Arctic Oscillation, the Pacifid Decadal Oscillation, the Tropical North Atlantic index), together with three local time series located in Western Europe (Belgium), are selected. The analysis is performed on time scales from a month to 5 years by using a sliding window as filtering procedure. A few key new results on the remote influence emerge: (i) The Arctic Oscillation plays a key role at short time (monthly) scales on the dynamics of the North Pacific and North Atlantic; (ii) the North Atlantic Oscillation is playing a global role at long time scales (several years); (iii) the Pacific Decadal Oscillation is indeed slaved to other influences; (iv) the local observables over Western Europe influence the variability on the ocean basins on long time scales. These results further illustrate the power of the Liang-Kleeman information flow in disentangling the dynamical dependencies.

@article{doi1016993tellusa44,
    author = "Vannitsem, Stéphane and Liang, X. San",
    title = "Dynamical Dependencies at Monthly and Interannual Time Scales in the Climate System: Study of the North Pacific and Atlantic Regions",
    year = "2022",
    journal = "Tellus A Dynamic Meteorology and Oceanography",
    abstract = "The directional dependencies of different climate indices are explored using the Liang-Kleeman information flow in order to disentangle the influence of certain regions over the globe on the development of low-frequency variability of others. Seven key indices (the sea-surface temperature in El-Niño 3.4 region, the Atlantic Multidecadal Oscillation, the North Atlantic Oscillation, the North Pacific America pattern, the Arctic Oscillation, the Pacifid Decadal Oscillation, the Tropical North Atlantic index), together with three local time series located in Western Europe (Belgium), are selected. The analysis is performed on time scales from a month to 5 years by using a sliding window as filtering procedure. A few key new results on the remote influence emerge: (i) The Arctic Oscillation plays a key role at short time (monthly) scales on the dynamics of the North Pacific and North Atlantic; (ii) the North Atlantic Oscillation is playing a global role at long time scales (several years); (iii) the Pacific Decadal Oscillation is indeed slaved to other influences; (iv) the local observables over Western Europe influence the variability on the ocean basins on long time scales. These results further illustrate the power of the Liang-Kleeman information flow in disentangling the dynamical dependencies.",
    url = "https://doi.org/10.16993/tellusa.44",
    doi = "10.16993/tellusa.44",
    openalex = "W4293146516",
    references = "doi10106315019944"
}

Causality is the relationship between causes and effects. Following Relativity, any cause of an event must always be in the past light cone of the event itself, but causes and effects must always be related to some interactions. In this paper, causality is developed as a consequence of the analysis of the Einstein, Podolsky, and Rosen paradox. Causality is interpreted as the result of time generation, due to irreversible interactions of real systems among them. Time results as a consequence of irreversibility; so, any state function of a system in its space cone, when affected by an interaction with an observer, moves into a light cone or within it, with the consequence that any cause must precede its effect in a common light cone.

@article{doi103390math10152711,
    author = "Lucia, Umberto and Grisolia, Giulia",
    title = "Thermodynamic Definition of Time: Considerations on the EPR Paradox",
    year = "2022",
    journal = "Mathematics",
    abstract = "Causality is the relationship between causes and effects. Following Relativity, any cause of an event must always be in the past light cone of the event itself, but causes and effects must always be related to some interactions. In this paper, causality is developed as a consequence of the analysis of the Einstein, Podolsky, and Rosen paradox. Causality is interpreted as the result of time generation, due to irreversible interactions of real systems among them. Time results as a consequence of irreversibility; so, any state function of a system in its space cone, when affected by an interaction with an observer, moves into a light cone or within it, with the consequence that any cause must precede its effect in a common light cone.",
    url = "https://doi.org/10.3390/math10152711",
    doi = "10.3390/math10152711",
    openalex = "W4289173373",
    references = "doi1010029780470432709, doi101002andp19053221004, doi101002andp19273892002, doi1010079789401708494, doi101007bf01449770, doi101007bf01491891, doi101016jppnp2020103812, doi101038121580a0, doi10106313051743, doi101103physrev32858, doi101103physrev47777, donoghue2020quantum"
}

We basis our initial analysis of the arrow of time on a relationship between the time evolution operator of quantum system and the time-independent density operator which describes the equilibrium state of a many-particle system at temperature $T$. We highlight through this analysis the identification of the imaginary temporal component of the branch-cut complex cosmic form factor with the direction in which the time-parameter flows globally, or the arrow of time. As a novelty, in this work we calculate the number of branches in the branch-cut universe to achieve causality involving the global time of evolution of the universe and the local time of travel of the light around each Hubble horizon. The preliminary result obtained is comparable to 60 e-folds of contraction in the FLRW cosmic scale factor $a(t) $ to overcome causality achieved in the bouncing model.

@misc{doi1048550arxiv221202670,
    author = "Bodmann, B. and Vasconcellos, César A. Zen and de Freitas Pacheco, José and Hess, Peter O. and Hadjimichef, Dimiter",
    title = "Causality and the Arrow of Time in the Branch-Cut Cosmology",
    year = "2022",
    booktitle = "arXiv (Cornell University)",
    abstract = "We basis our initial analysis of the arrow of time on a relationship between the time evolution operator of quantum system and the time-independent density operator which describes the equilibrium state of a many-particle system at temperature $T$. We highlight through this analysis the identification of the imaginary temporal component of the branch-cut complex cosmic form factor with the direction in which the time-parameter flows globally, or the arrow of time. As a novelty, in this work we calculate the number of branches in the branch-cut universe to achieve causality involving the global time of evolution of the universe and the local time of travel of the light around each Hubble horizon. The preliminary result obtained is comparable to 60 e-folds of contraction in the FLRW cosmic scale factor $a(t) $ to overcome causality achieved in the bouncing model.",
    url = "https://doi.org/10.48550/arxiv.2212.02670",
    doi = "10.48550/arxiv.2212.02670",
    openalex = "W4310881808"
}

We argue that the asymmetry between diverging and converging electromagnetic waves is just one of many asymmetries in observed phenomena that can be explained by a past hypothesis and statistical postulate (together assigning probabilities to different states of matter and field in the early universe). The arrow of electromagnetic radiation is thus absorbed into a broader account of temporal asymmetries in nature. We give an accessible introduction to the problem of explaining the arrow of radiation and compare our preferred strategy for explaining the arrow to three alternatives: (i) modifying the laws of electromagnetism by adding a radiation condition requiring that electromagnetic fields always be attributable to past sources, (ii) removing electromagnetic fields and having particles interact directly with one another through retarded action-at-a-distance, (iii) adopting the Wheeler-Feynman approach and having particles interact directly through half-retarded half-advanced action-at-a-distance. In addition to the asymmetry between diverging and converging waves, we also consider the related asymmetry of radiation reaction.

@article{doi101016jshpsa202301002,
    author = "Hubert, Mario and Sebens, Charles T.",
    title = "Absorbing the arrow of electromagnetic radiation",
    year = "2023",
    journal = "Studies in History and Philosophy of Science Part A",
    abstract = "We argue that the asymmetry between diverging and converging electromagnetic waves is just one of many asymmetries in observed phenomena that can be explained by a past hypothesis and statistical postulate (together assigning probabilities to different states of matter and field in the early universe). The arrow of electromagnetic radiation is thus absorbed into a broader account of temporal asymmetries in nature. We give an accessible introduction to the problem of explaining the arrow of radiation and compare our preferred strategy for explaining the arrow to three alternatives: (i) modifying the laws of electromagnetism by adding a radiation condition requiring that electromagnetic fields always be attributable to past sources, (ii) removing electromagnetic fields and having particles interact directly with one another through retarded action-at-a-distance, (iii) adopting the Wheeler-Feynman approach and having particles interact directly through half-retarded half-advanced action-at-a-distance. In addition to the asymmetry between diverging and converging waves, we also consider the related asymmetry of radiation reaction.",
    url = "https://doi.org/10.1016/j.shpsa.2023.01.002",
    doi = "10.1016/j.shpsa.2023.01.002",
    openalex = "W4360884810",
    references = "doi101016s1355219899000301"
}

@article{doi101140epjss11734023009792,
    author = "Plotnitsky, Arkady",
    title = "t is not time: reality, causality, and the arrow of events in quantum theory",
    year = "2023",
    journal = "The European Physical Journal Special Topics",
    url = "https://doi.org/10.1140/epjs/s11734-023-00979-2",
    doi = "10.1140/epjs/s11734-023-00979-2",
    openalex = "W4386476348",
    references = "doi101001archinte195900270050166032, doi1010160003491657900325, doi101016016093278490022x, doi101038nphys2930, doi101088003191121011015, doi101103physrev48696, doi10111914874855, doi1015159783110213195, doi1043249780203132241, doi105860choice380352"
}

Academic interest in well-being has blossomed, to the point that numerous forms of well-being have been proposed, covering myriad aspects of the person (e.g. mental, physical, social, spiritual) and of life more broadly (e.g. communal, economic, environmental). This proliferation of forms raises the question of how they might ideally interrelate, and whether there is some kind of overall well-being that draws them together. To that end, this paper argues that a zenith of ultimate or complete well-being would involve managing to sustain well-being across numerous systems (i.e. configurations of different processes and entities), such that they are in balance and harmony. These systems include: (a) the various dimensions of the person; (b) self-and-other (c) people-and-environment; and (d) time. We suggest that attaining all these various forms of sustainable well-being constitutes an ideal of flourishing to which people and societies can and should aspire.

@article{doi1010801743976020242362435,
    author = "Lomas, Tim and Pawelski, James O. and VanderWeele, Tyler J.",
    title = "Flourishing as ‘sustainable well-being’: Balance and harmony within and across people, ecosystems, and time",
    year = "2024",
    journal = "The Journal of Positive Psychology",
    abstract = "Academic interest in well-being has blossomed, to the point that numerous forms of well-being have been proposed, covering myriad aspects of the person (e.g. mental, physical, social, spiritual) and of life more broadly (e.g. communal, economic, environmental). This proliferation of forms raises the question of how they might ideally interrelate, and whether there is some kind of overall well-being that draws them together. To that end, this paper argues that a zenith of ultimate or complete well-being would involve managing to sustain well-being across numerous systems (i.e. configurations of different processes and entities), such that they are in balance and harmony. These systems include: (a) the various dimensions of the person; (b) self-and-other (c) people-and-environment; and (d) time. We suggest that attaining all these various forms of sustainable well-being constitutes an ideal of flourishing to which people and societies can and should aspire.",
    url = "https://doi.org/10.1080/17439760.2024.2362435",
    doi = "10.1080/17439760.2024.2362435",
    openalex = "W4399298986",
    references = "doi1010801743976020222131608"
}

In the light of almost a century of struggle to make (common) sense of Quantum Mechanics and to reconcile it with General Relativity, it is proposed to (for some time) forget about quantizing gravity or striving for one Theory of Everything or “Weltformel”, which would describe the whole of reality without any joints or suture marks. Instead of one single monolithic formalism, a three-legged compound approach is argued for. Quantum Mechanics, Relativity and Thermodynamics are proposed as the main pillars of reality, each with its well-defined realm, specific features, and clearly marked interfaces between the three of them. Not only classical reality, which is rather directly accessible to us, is then comprehensively modelled by their encompassing combination. Quantum phenomena are understood as undoubtedly lying at the bottom of classical physics and at the same time, they become “fully real” only when embedded in classical frames between preparation and measurement in time. It is then where thermodynamics steps in and provides the mediating glue. The aim of this short contribution is not to deliver novel quantitative results but rather to propose a comprehensive research program and to coarsely lay out a very roughly coherent self-referential sketch starting from the beginning of the one universe, which we inhabit.

@misc{doi1020944preprints2024031778v1,
    author = "Thomsen, K.",
    title = "A Heuristic Sketch How It Could Fit All Together With Time",
    year = "2024",
    booktitle = "Preprints.org",
    abstract = "In the light of almost a century of struggle to make (common) sense of Quantum Mechanics and to reconcile it with General Relativity, it is proposed to (for some time) forget about quantizing gravity or striving for one Theory of Everything or “Weltformel”, which would describe the whole of reality without any joints or suture marks. Instead of one single monolithic formalism, a three-legged compound approach is argued for. Quantum Mechanics, Relativity and Thermodynamics are proposed as the main pillars of reality, each with its well-defined realm, specific features, and clearly marked interfaces between the three of them. Not only classical reality, which is rather directly accessible to us, is then comprehensively modelled by their encompassing combination. Quantum phenomena are understood as undoubtedly lying at the bottom of classical physics and at the same time, they become “fully real” only when embedded in classical frames between preparation and measurement in time. It is then where thermodynamics steps in and provides the mediating glue. The aim of this short contribution is not to deliver novel quantitative results but rather to propose a comprehensive research program and to coarsely lay out a very roughly coherent self-referential sketch starting from the beginning of the one universe, which we inhabit.",
    url = "https://doi.org/10.20944/preprints202403.1778.v1",
    doi = "10.20944/preprints202403.1778.v1",
    openalex = "W4393867694",
    references = "doi103390math10152711"
}

@incollection{crossref2025causality,
    title = "Causality, Anti-causality and Machine Learning",
    year = "2025",
    booktitle = "Fractional Calculus",
    url = "https://doi.org/10.1142/9789819814404\_0009",
    doi = "10.1142/9789819814404\_0009",
    pages = "135-157"
}

On a scientific meta-level, it is discussed how an overall understanding of the physical universe can be built on the basis of well-proven theories, observations, and recent experiments. In the light of almost a century of struggle to make (common) sense of Quantum Mechanics and to reconcile it with General Relativity, it is proposed to (for some time) forget about quantizing gravity or striving for one Theory of Everything or “Weltformel”, which would describe the whole of reality seamlessly without any joints or suture marks. Instead of one single monolithic formalism, a three-legged compound approach is argued for. Quantum Mechanics, Relativity and Thermodynamics are proposed as the main pillars of reality, each with its well-defined realm, specific features, and clearly marked interfaces between the three of them. Not only classical reality, which is rather directly accessible to us, is then comprehensively modelled by their encompassing combination. Quantum phenomena are understood as undoubtedly lying at the bottom of classical physics and at the same time, they become “fully real” only when embedded in classical frames, i.e., preparation and measurements in time. It is then where thermodynamics steps in and provides the mediating glue as it does at interfaces towards gravity. Decoherence is understood as a smooth way of gradually transferring information and basically dumping entropy to a suitable environment. The aim of this short contribution is not to deliver novel quantitative results but rather to propose a comprehensive research program and to coarsely lay out a very roughly coherent sketch starting from the beginning of the one universe, which we inhabit. The all-embracing picture is claimed to be one of (“mutually induced”) emergence.

@article{doi1032388vnfm11,
    author = "Thomsen, K.",
    title = "A Heuristic Sketch of How It Could All Fit Together with Time",
    year = "2025",
    journal = "Qeios",
    abstract = "On a scientific meta-level, it is discussed how an overall understanding of the physical universe can be built on the basis of well-proven theories, observations, and recent experiments. In the light of almost a century of struggle to make (common) sense of Quantum Mechanics and to reconcile it with General Relativity, it is proposed to (for some time) forget about quantizing gravity or striving for one Theory of Everything or “Weltformel”, which would describe the whole of reality seamlessly without any joints or suture marks. Instead of one single monolithic formalism, a three-legged compound approach is argued for. Quantum Mechanics, Relativity and Thermodynamics are proposed as the main pillars of reality, each with its well-defined realm, specific features, and clearly marked interfaces between the three of them. Not only classical reality, which is rather directly accessible to us, is then comprehensively modelled by their encompassing combination. Quantum phenomena are understood as undoubtedly lying at the bottom of classical physics and at the same time, they become “fully real” only when embedded in classical frames, i.e., preparation and measurements in time. It is then where thermodynamics steps in and provides the mediating glue as it does at interfaces towards gravity. Decoherence is understood as a smooth way of gradually transferring information and basically dumping entropy to a suitable environment. The aim of this short contribution is not to deliver novel quantitative results but rather to propose a comprehensive research program and to coarsely lay out a very roughly coherent sketch starting from the beginning of the one universe, which we inhabit. The all-embracing picture is claimed to be one of (“mutually induced”) emergence.",
    url = "https://doi.org/10.32388/vnfm11",
    doi = "10.32388/vnfm11",
    openalex = "W4408298266",
    references = "doi103390math10152711"
}

@incollection{fowlerNonecausality,
    author = "Fowler, O. S.",
    title = "Causality.",
    year = "None",
    booktitle = "Memory and intellectual improvement applied to self-education and juvenile instruction (25th ed., improved).",
    url = "https://doi.org/10.1037/12006-014",
    doi = "10.1037/12006-014",
    pages = "168-188"
}